Does this happen a lot?

What? The texture loss in the background or the engine failure? 

Police cars having engine failures.
I may have too many vehicle mods.

It's not entirely about the vehicle mods, it's dependant on how well made they are, some mods when spawned multiple times result in texture loss. SO you might want to go through each vehicle to find which one causes it. Just replace it and then you should be able to have as many mods as you want (within reason based on the performance of your PC). With regard to the engine, to be honest I've never had that before, ever. Unless it's done by the Simple Native trainer or you crash into every single thing, I've never had the engine failures before. 

that cop spawned with the broken down car
right now I'm not going to worry about it, as I have no time to play gta 4