Having learn how to drive in Europe I find intersection particularly tricky in NY state in USA.

NY City is particularly tricky, and they get exceptions from national model laws. For instance, "right on red light (if traffic clear)" is forbidden in NYC, and that is taught in many states' driver's education manuals that are nowhere near NY state...

I was often faced with the following situation in a intersection when I was not sure how to proceed:

I am in the left lane and I want to do a left turn (there a simple green light but no arrow sign, but also no prohibition of left turns), I yield to opposite direction traffic, but then after the left turn when entering in the perpendicular road, what can you do if there is a crosswalk with a light green for the pedestrian, and there are no pedestrian passing?

So you are finishing your turn and there are no pedestrians in your way. You notice that the pedestrian path is signaled for pedestrians to use (white guy walking light with perhaps number counting down). That signal is not for you. You got your signal earlier. Your job here is to assure you don't hit any pedestrians, and don't even come close. So don't try to time it so you whoosh past while they've barely cleared by 1m or whatever. Wide berth!

Let's have a word about what a crosswalk is.

In America, a crosswalk is the imaginary* line extending along edges (would-be sidewalks) of any cross streets. So on a common + intersection with 2 stop signs and NO pavement markings, there are 4 crosswalks.

When a minor road T's into a major road, that creates 3 crosswalks.

Crosswalks are also, obviously anywhere marked as a crosswalk.

But the imaginary crosswalks have the same legal force as the marked ones.

So again, if you have a 5-way -- say a Y and - shape superimposed -- and there are obvious crosswalk markings on only 2 of the five sides... you have how many crosswalks? Anyone? Anyone? Bueller?

FIVE crosswalks. Three of which are invisible.

Now, the only way intersections do not cast invisible crosswalks, is where there are prohibitive devices to forbid it. Those are aimed at pedestrians; you can be aware of them but expect chaotic behavior.

And of course, freeways do not have crosswalks.

I am in the right lane and I want to make a right turn on red when there is a crosswalk light with a green for pedestrian (with no pedestrian passing).

Again, that signal is not for you. You must use all the normal diligence when doing a Right On Red, including not hitting pedestrians. In fact, if the signal is against the pedestrian, you still must watch for them!

When you see pedestrian devices (like the signal, or simply like a marked crosswalk), you as a driver know this is a high traffic/risk pedestrian crossing, and give it extra attention.

New York State FAQ concerning drivers and pedestrians:

FAQs & Laws

Pedestrians

• What’s the best way to use a pedestrian traffic signal?

  A pedestrian has the right of way when the pedestrian signal shows a steady “Walk” sign or person symbol.

  A pedestrian should not cross the road if a “Don’t Walk” sign or upraised hand symbol is steady.

  When a “Don’t Walk” sign or upraised hand symbol is flashing, pedestrians who have already begun crossing the street should continue to the other side. Pedestrians who have not yet begun to cross the street should wait until the next “Walk” cycle.

• What is considered a crosswalk?

  A crosswalk is any area distinctly indicated for pedestrian crossing by lines or other pavement markings.

  A crosswalk also is any part of the road at an intersection between the curbs on opposite sides of the roadway or, if there are no curbs, between the edges of the road. This area is considered a crosswalk, whether or not it is marked as one.

• How should a crosswalk be used?

  When there are no pedestrian signals at an intersection, wait until vehicle traffic gets a green light in the same direction you are traveling and cross in front of the stopped traffic. Do not cross in front of traffic that has a green light.

  If there is a crosswalk at a location where there are no traffic control signals for drivers or pedestrians, drivers must yield the right of way to pedestrians.

• What if there isn’t a marked crosswalk?

  If there are no crosswalks, the safest place for pedestrians to cross the road is at an intersection. Motorists have the right of way at all locations other than intersections and marked crosswalks.

• Are pedestrians required to use a sidewalk if one is provided?

  Yes. Where sidewalks are provided and they can be used safely, pedestrians must use the sidewalk instead of walking in the road.

• Who has the right of way when a car is pulling into or out of traffic across a sidewalk?

  Pedestrians do. The driver of a vehicle emerging from or entering an alleyway, building, private road or driveway must yield the right of way to any pedestrian approaching on any sidewalk or road shoulder.

• If no sidewalk is provided, should a pedestrian walk on a road shoulder facing oncoming traffic or with their back to oncoming traffic?

  Pedestrians should walk on the shoulder facing oncoming traffic because it makes them more visible to motorists and helps them stay aware of traffic. Stay as far to the left as possible.

• Are pedestrians allowed on state expressway highways or state interstate route highways including entrance ramps and exit ramps?

  No. Pedestrians are not allowed on limited-access roads, such as expressways and interstates, nor are they allowed on entrance or exit ramps for those roads.

• How should pedestrians cross the road at roundabouts?

  Pedestrians should cross roundabouts from one “splitter island” to the next. The splitter islands offer a safe refuge between the two different directions of traffic, allowing a pedestrian to concentrate on traffic coming from only one direction at a time. Never cross to the center island of a roundabout.

Drivers

• When do pedestrians have the right of way?

  Pedestrians have the right of way in all crosswalks and at intersections with marked or unmarked crosswalks.

  If an intersection is equipped with a pedestrian traffic signal, they should cross during the “Walk” phase of the signal.

  *Note: a pedestrian will get a “Walk” signal to go the same direction motorists are heading when they get a green signal. Motorists turning right or left at an intersection should always look for pedestrians and yield the right of way to them. Pedestrians have the right of way at intersections, even if drivers have a green light.

• Who has the right of way when I’m entering or exiting the road?

  Pedestrians do. The driver of a vehicle emerging from or entering an alleyway, building, private road or driveway shall yield the right of way to any pedestrian approaching on any sidewalk or road shoulder.

• What is considered a crosswalk?

  A crosswalk is any area distinctly indicated for pedestrian crossing by lines or other pavement markings.

  A crosswalk also is any part of the road at an intersection between the curbs on opposite sides of the roadway or, if there are no curbs, between the edges of the road. This area is considered a crosswalk whether or not it is marked.

SECTIONS OF THE VEHICLE & TRAFFIC LAW PERTAINING TO PEDESTRIANS

DEFINITIONS:

Section 110. Crosswalk.
(a) That part of a roadway at an intersection included within the connections of the lateral lines of the sidewalks on opposite sides of the highway between the curbs or, in the absence of curbs, between the edges of the traversable roadway.
(b) Any portion of a roadway at an intersection or elsewhere distinctly indicated for pedestrian crossing by lines or other markings on the surface.

Section 130. Pedestrian. Any person afoot or in a wheelchair.

Section 1112. Pedestrian-control signal indications. Whenever pedestrians are controlled by pedestrian-control signals exhibiting the words “WALK” or “DON’T WALK”, or exhibiting symbols of a walking person or upraised hand, such signals shall indicate and apply to pedestrians as follows:
(a) Steady WALK or walking person. Pedestrians facing such signal may proceed across the roadway in the direction of the signal and shall be given the right of way by other traffic.
(b) Flashing DON’T WALK or upraised hand. No pedestrian shall start to cross the roadway in the direction of such signal, but any pedestrians who have partially completed their crossing on the WALK or walking person signal shall proceed to a sidewalk or safety island while the flashing DON’T WALK or upraised hand signal is showing.
(c) Steady DON’T WALK or upraised hand. No pedestrians shall start to cross the roadway in the direction of such signal, but any pedestrians who have partially completed their crossing on the WALK or flashing DON’T WALK signal shall proceed to a sidewalk or safety island while the steady DON’T WALK signal is showing.

INTERSECTIONS

Section 1142. Vehicle entering stop or yield intersection.
(b) The driver of a vehicle approaching a yield sign shall in obedience to such sign slow down to a speed reasonable for existing conditions, or shall stop if necessary as provided in section eleven hundred seventy-two, and shall yield the right of way to any pedestrian legally crossing the roadway on which he is driving, and to any vehicle in the intersection or approaching on another highway so closely as to constitute an immediate hazard during the time such driver is moving across or within the intersection. Provided, however, that if such driver is involved in a collision with a pedestrian in a crosswalk or a vehicle in the intersection after driving past a yield sign without stopping, such collision shall be deemed prima facie evidence of his failure to yield the right of way.

DUE CARE

Section 1146. Drivers to exercise due care. Notwithstanding the provisions of any other law to the contrary, every driver of a vehicle shall exercise due care to avoid colliding with any bicyclist, pedestrian or domestic animal upon any roadway and shall give warning by sounding the horn when necessary.

PEDESTRIANS' RIGHTS AND DUTIES ARTICLE 27

Section 1150. Pedestrians subject to traffic regulations. Pedestrians shall be subject to traffic-control signals as provided in section eleven hundred eleven of this title, but at all other places pedestrians shall be accorded the privileges and shall be subject to the restrictions stated in this article.

Section 1151. Pedestrians' right of way in crosswalks.
(a) When traffic-control signals are not in place or not in operation the driver of a vehicle shall yield the right of way, slowing down or stopping if need be to so yield, to a pedestrian crossing the roadway within a crosswalk on the roadway upon which the vehicle is traveling, except that any pedestrian crossing a roadway at a point where a pedestrian tunnel or overpass has been provided shall yield the right of way to all vehicles.
(b) No pedestrian shall suddenly leave a curb or other place of safety and walk or run into the path of a vehicle which is so close that it is impractical for the driver to yield.
(c) Whenever any vehicle is stopped at a marked crosswalk or at any unmarked crosswalk at an intersection to permit a pedestrian to cross the roadway, the driver of any other vehicle approaching from the rear shall not overtake and pass such stopped vehicle.

Section 1151-a. Pedestrians' right of way on sidewalks. The driver of a vehicle emerging from or entering an alleyway, building, private road or driveway shall yield the right of way to any pedestrian approaching on any sidewalk extending across such alleyway, building entrance, road or driveway.

Section 1152. Crossing at other than crosswalks.
(a) Every pedestrian crossing a roadway at any point other than within a marked crosswalk or within an unmarked crosswalk at an intersection shall yield the right of way to all vehicles upon the roadway.
(b) Any pedestrian crossing a roadway at a point where a pedestrian tunnel or overhead pedestrian crossing has been provided shall yield the right of way to all vehicles upon the roadway.
(c) No pedestrian shall cross a roadway intersection diagonally unless authorized by official traffic-control devices; and, when authorized to cross diagonally, pedestrians shall cross only in accordance with the official traffic-control devices pertaining to such crossing movements.

Section 1153. Provisions relating to blind or visually impaired persons.
(a) Notwithstanding the foregoing provisions of this article, every driver of a vehicle approaching an intersection or crosswalk shall yield the right of way to a pedestrian crossing or attempting to cross the roadway when such pedestrian is accompanied by a guide dog or using a cane which is metallic or white in color or white with a red tip.
(b) No person, unless blind or visually impaired, shall use on any street or highway a cane which is metallic or white in color or white with a red tip.
(c) This section shall not be construed as making obligatory the employment of the use of a guide dog or of a cane or walking stick of any kind by a person blind or visually impaired.

Section 1155. Pedestrians to use right half of crosswalks. Pedestrians shall move, whenever practicable, upon the right half of crosswalks.

Section 1156. Pedestrians on roadways.
(a) Where sidewalks are provided and they may be used with safety it shall be unlawful for any pedestrian to walk along and upon an adjacent roadway.
(b) Where sidewalks are not provided any pedestrian walking along and upon a highway shall when practicable walk only on the left side of the roadway or its shoulder facing traffic which may approach from the opposite direction. Upon the approach of any vehicle from the opposite direction, such pedestrian shall move as far to the left as is practicable.

Section 1157. Pedestrians soliciting rides, or business.
(a) No person shall stand in a roadway for the purpose of soliciting a ride, or to solicit from or sell to an occupant of any vehicle.
(b) No person shall stand on or in proximity to a street or highway for the purpose of soliciting the watching or guarding of any vehicle while parked or about to be parked on a street or highway.
(c) No person shall occupy any part of a state highway, except in a city or village, in any manner for the purpose of selling or soliciting.