You don't need an asi loader or scripthook for vehicle mods, but simple native trainer might be usefull to spawn things in multiplayer.
Car mods should work online, just make sure you don't install any car mods that change structural things about the car. For instance I once installed a truck that had a longer loading space in which you could stand. When spawning that truck in multiplayer and getting in the loading space I got the SMPA-60 error. It has something to do with conflicting files (you're standing somewhere you're technically not suppose to stand according to other people's games).
What do you mean by the car not being there? Not at the location it usually spawns? Make sure you set your map to the entire liberty city when setting up your server.
Like I said, might be useful to use simple native trainer , so that you can just spawn the vehicle you replaced. I recently uploaded all the required files on mediafire for someone so I'll just give you that too (I have the same patch as you).
It has dsound.dll as asiloader so you'll still have the games for windows live interface, and you'll still be able to play online. With this trainer you can use F3 ingame to open it up, and use numpad to browse through it. It's really handy for getting all weapons, spawning vehicles etc.
For your information, you will be placed on different servers then people who don't have mods. To prevent people from cheating in regular servers. Doesn't matter how many mods or what kind you got, 1 is enough to place you on those.