Where can I leave luggage in Berlin and be able to pick it late at night?

To use the lockers in a train station is a good idea, however if your flight is in Schönefeld you should use the lockers at Südkreuz station. As in all stations the cost is 4 Euros for a small locker, 6 Euros for a large locker (per 24 hours).

(BTW. if this is a stopover and you also arrive at SFX, then Schönefeld Airport has lockers, so the simplest option would be to leave your luggage there.)

The lockers as Südkreuz are located at the exit Hildegard-Knef-Platz. They are inside the building, and the police station inside the station is just around the corner, so your luggage should be pretty safe (take care not to leave any luggage unattended in the open or it will be removed and destroyed).

Südkreuz is also a good option because it is a traffic hub.

It's probably worth buying a "Tageskarte Berlin ABC" (day ticket for all fare zones) at 7.5 Euro (valid until 3.00 am of the next day).

From Südkreuz you can either use the S42 train (going to Ostkreuz). This will take you to the districts of Kreuzberg, Friedrichshain and Prenzlauer Berg (more or less the "party districts" of Berlin). The S42 is part of the "Ring", the line that goes in circles around the inner city (if you imagine a clock face with the most northern station (Gesundbrunnen) at 12 o'clock the S42 goes counter-clockwise and the S41 train goes clockwise. A full circle takes more than an hour and the view is not worth it, so try to catch the right direction).

However the "Ring" touches Kreuzberg (probably the most famous of Berlin's districts) only at the periphery, if you want to go there you might want to get off the S42 at Hermannstrasse and proceed with the tube U8 to Kottbusser Tor, right in the middle of Kreuzberg.

At Südkeuz you can also take the S2 train going to Friedrichstrasse. If you want to see a few sights you get get off at Potsdamer Platz.

Potsdamer Platz itself is more family-oriented entertainment - shopping malls, restaurants, cinemas, the film museum, waxworks and Legoland. But you can proceed by foot along the "Kulturforum" (if you interested in architecture; the "Philharmonie", the most famous Berlin concert hall, is an interesting albeit not beautiful sight), stroll by the Holocaust Memorial until you arrive at the parliament building (the former Reichstag). From there you can stroll down the avenue Unter den Linden to Friedrichstrasse and return with the S2 to Südkreuz.

Südkreuz station is located in the Schöneberg district, so you could also explore that. The area around the station is only just being developed, there is not much to see yet. You'd have to take the 204 bus to Kaiser-Wilhelm-Platz (the day ticket is good for buses too).

For a selection of bars and cafes you can enter Crellestrasse just right the corner of Kaiser Wilhelm-Platz, or ask for Akazienstrasse (Akazienstrasse and the streets that branch off have a lot of bars, restaurants, clubs etc). Schöneberg has also a vivid gay scene, if you specifically interested in / specifically want to avoid this you might want to go to / avoid Bülowstrasse and surroundings.

From Südkreuz station the S45 will bring you to Schönefeld Airport (goes every 30 minutes). However the last direct connection is about at 22.45, which is probably too early for you - there are a few other options (S42 to Treptower Park, then change to S9 to Schönefeld), but past midnight you will probably have to go by bus. S45 resumes operations at about 4.45 am.

Since you were talking about skipping the night at a hotel, it would be one convenient solution for you to look for a room in Schöneberg (not many hotels, but a lot of B&Bs and the like) near Südkreuz and explore the city from there.