How did the Hansa organize river trade in the 14th and 15th centuries?

I believe that after a lot of research I can start the basis of an answer. My understanding is that the answer to your question, as @Steve Bird said, would depend on the river. There have been found wrecks of cogs in rivers, most notable among them the Bremen cog. Bremen is a city upstream in the Weser river and had a shipyard. The cog, as I said in my comment, was able to travel up river, especially in rivers found in many of the most important german Hanseatic cities, and maybe even played a major role in their development as attested here:

The architecture of the Cog informs what kinds of harbors and ports can be developed as trade centers develop in new locations. Because of the morphology of the bottom of the hull, Cogs are able to travel on rivers to inland harbors such as those found in Bremen and Lübeck and do not require the coastal harbors that ships with tapered bottoms did. However, Cogs, once beached, were extremely difficult to get back to sea, unlike Keels or other tapered bottom ships, and therefore required docks and mooring points to be constructed in the harbors for the ships. Additionally, the architecture of the Cog reflects its primary function as a commercial shipping vessel.

However, other river cities, had different problems. Another example is Kampen:

Kampen was the largest Hanseatic seaport in the Low Countries.. Goods were transferred here from river barges to sea going vessels (and the other way around) connecting the Rhineland with trading centres in the Baltic such as Lubeck and Gdansk, Scandinavia and England. These routes took the ships on voyages across the dangerous shallows of the Zuiderzee and Waddenzee, towards the North Sea.

And another notable example is Ladoga, as told in Baltic Hospitality from the Middle Ages to the Twentieth Century:

Not far from Ladoga, upstream the Volkhov River, there were the most dangerous of the Eastern European rapids, the Volkhovskie (Gostinopol’skie) rapids. They stretched for 9 km between the steep limestone banks that were over 20 m high. Thirty kilometers farther downstream began the other rapids—these ones known as the Pchevskie—which stretched for another 9 km and created further difficulties for navigation. Overcoming the rapids required travelers to reload goods, transporting them overland along the river, and dragging or pulling ships on ropes along the coast. The conditions for crossing the rapids are described in detail in the 1269 treaty of Novgorod with Lübeck and Gotland. The ships on which the Germans brought their goods to the Novgorodian Land could not pass the Volkhov rapids. The goods were reloaded onto flat-bottomed Novgorodian boats that were led to Il’men’ by local pilots. Even experienced pilots were not safe when passing the rapids, as evidenced by a clear division of responsibility: the German guests were responsible for the ship itself, whereas the pilots were not responsible for the sunken goods.

Who owned and facilitated that transportation of goods up the river? In the case of the Volkhov rapids, the boats were Novgorodian. I do not have yet a definitive answer but a good guess would be that this was either regulated by agreements between the Hansa, the local kontor and the city's authorities. If that wasn't an option they should be able to hire boats. Or a merchant of the city could transport the goods for them. The options were many and can be seen (and admired for how advanced they were for their time) here:

1. The "vera societas" (wedderlegginge) was a specific commercial enterprise in which the financier (Kapitalgeber) and the person responsible for the commercial transaction (Kapitalführer) shared in the profits.

2. In the case of a "commission business" (sendeve), a merchant transported goods for another merchant. Usually, the two merchants would also have entered into a "vera societas" with each other.

3. The "trading company" consisted of two or more persons, who shared in the company's capital, but were themselves also involved in active trading. This type of business developed at the beginning of the 15th century.

4. The "reciprocal trade" was the most important form of partnership between the Hanse merchants. In this instance, merchants in different places would sell the respective commodities sent to them by other merchants in their own names. For this practice there were, however, no written agreements, and the sellers did not share directly in the profits of a particular commercial transaction.26

5. For the organization of long-distance travel, the merchants formed "cooperatives", for example those operating in Scania, Bergen or Flanders.

From the source, I can answer your 2nd additional question and say that frozen rivers would not be an issue as travelling during winter seems to have been prohibited:

To reduce the risk of accident, winter travel was prohibited on the Baltic Sea. However, because the forced winter residence led to increased costs in foreign ports, this policy was disadvantageous for the Hanse merchants in the long run. What is more, the non-Hanse merchants continued with their maritime trade during the winter.

So to recap my answer, river trade was as complex as the rivers the Hanseatic traders used. How they conducted their river trade was subject to the morphology of the river and agreements with the interested cities. The loose structure of the trade network meant that river transactions, as all of their transactions would have been concluded in any number of ways.