Car lock cover plate jammed in lock

You can probably retrieve that piece that jammed in there using a thin wire bent into a hook shape. Better would be lockpicking tools but those may not be available to you, a thin paper clip should work okay. You would work it into the back of the lock and then pull the debris out, as soon as a bit pokes out you grab it with a pliers and get it the rest of the way. I would do this to get myself driving again.

As to where to take it for a more permanent repair you'd take it to a mechanic, not a locksmith, unless that locksmith specializes in cars. The doors need to be opened up so the lock cylinders can be removed and replaced. While they are at it they could fix the central locking system so your fob works again. As to what you should do that is a matter of opinion and how much the car is worth for resale. Having obvious problems like broken central locking would generally make me walk away, or at least reduce the value I'd pay for it. If you have a common make where there's lots of choice having broken things on the car will make it harder to sell, but if the value is very low it may cost more to fix these things than they are worth to the value.