The GTA 3 era games by themselves are not bad, it's just Vice City Stories that has a lot of glitches and physics problems. I love Vice City Stories, but I (and any other sensible VCS fan) will admit that it has a ton of glitches. It's why VCS is usually thought of as the worst 3D GTA game. However, I've played both the PSP and PS2 versions extensively, and I believe that you are less likely to experience glitches and frustration on the PS2 version.
Some of the glitches were fixed when the game was ported, and some of them are just negated since you have more control over the game with the double analog sticks for camera aim, and the R2/L2 buttons for switching between targets, for example.
I've experienced the glitch with no cars on the street quite a few times, mostly on the PSP version. I assume that's what you are playing. It always creeped me out as well. Sometimes the game would get very foggy or dark, and everyone and every vehicle would disappear... never figured it out, but probably just the PSP running out of memory and locking the game into an empty cycle or something.
Helicopters are real hard to control on the PSP. I've played GTA games on many platforms (PS2, Xbox 360, GBA, Nintendo DS, PC) and I've always found the PS2 controller to be the very best controller for piloting helicopters. That being said, while you probably know that this isn't your problem, my right shoulder button on my PSP was always getting stuck down, so it would crash my helicopters and stop my cars, and I had to pause the game and get it "unstuck"
. Was it raining when the helicopters messed up like this? I don't remember what direction it was in, but whenever there was a storm, the "winds" would always push helicopters in one direction and make them hard to fly.