I must say that while the game is really cool to play - even the sim part has its charm - the shooting sequence is just awful and I'm having trouble finishing it - I just keep dying !. Any way to bypass it quickly ? Some cheat code, trainer program or something like that ? Maybe slowing the game down in Dosbox will help ?

Don't know any cheats, but I did find shooting sequences relatively simple. Then again I used to play more challenging shoot 'em ups back in the 80s. Shooting sequences in Mean Streets are relatively simple. There's endless number of enemies and you must go to the other end of the room. Best tactic is to take cover and wait your chance, then stand up and shoot and quickly move little bit forward before next enemy starts shooting. Repeat ad infinitum.

Two little tricks:
1) Right after you shoot you can't run for a second, but you can duck. Pop up, shoot, duck, pop up and run, repeat.
2) If there's nothing between you and a baddie, you can shoot him in the gut while crouched, whereas they never fire while crouched. Occasionally useful.

I've also found that if you can get to a spot close to the edge of some cover, they wont be able to shoot you but you can shoot them.

Also, they appear based on how much time you spend on the screen rather than actual numbers, so when you're clear cover as much ground as you can and you'll have less of them to deal with.