What are the controls?
Press enter to choose a menu option, and escape to go back to the previous menu. Press F9 to view in-game statistics, and P to pause. Alt+enter switches between full-screen and windowed mode.

The other controls are configurable via the "setup options" menu. By default, the controls are as follows:
-Left and right arrow keys to move left and right.
-Under Windows, Alt Gr (right Alt) to jump and the space bar to shoot.
-Under Linux, the space bar to jump and left Alt to shoot.
-Right Ctrl to change weapon.
Does OpenJazz work with JCS'94/J1E levels?
It works with most of them. OpenJazz does not perfectly emulate all the quirks of the original game, though. These levels tend to use the engine in ways that were not originally intended. Consequently, it's not always possible to get to the end of new levels.

By the way, OpenJazz doesn't need to be in the same folder as your level files. You can keep them in another folder, and play them by passing the path on the command line. You can do this with as many folders as you like, so you don't need to replace any original levels with custom ones!
What port does OpenJazz use for online play?
Port 10052, just like Jazz Jackrabbit 2.
I start the program, and a window appears and then immediately disappears
The program produces a file called stdout.txt, which contains a list of the files OpenJazz has tried to read. Do they exist? OpenJazz requires the files from any of the following games: Jazz Jackrabbit, Jazz Jackrabbit CD, Jazz Jackrabbit Shareware Edition, Jazz Jackrabbit Christmas Edition, or Jazz Jackrabbit: Holiday Hare 1995. OpenJazz will not work with any of the Jazz Jackrabbit 2 games.
The physics don't feel right
Without knowing how the physics in the original game worked, it's impossible to get them exactly right.