1955: When Marty couldn't use the time circuits anymore was the car still actually driveable?

If you recall, the car wouldn’t start. That’s why he had to push it when he hid it behind the sign. But also cars in 1955 didn’t look anything like DeLoreans so he would have caused himself a lot of problems and ended up with a lot of questions - way worse than the puffy 80s vest jacket.

Remember, when old man Peabody sees the DeLorean in his barn after the bumpy entry into 1955, he says it looks like an airplane without wings. He doesn’t even see that it’s a car.

It has occurred to me I haven’t really answered the title question. The DeLorean that goes from 1985 to 1955 in the first movie runs on gasoline (we’ll ignore any potential leaded/unleaded questions). And we know from the climax of the movie when Marty drives through Hill Valley to reach 88 mph just as the lightning strikes the clock tower that the nuclear reaction is only needed for an actual trip through time. The time circuits, flux capacitor (in what I will call “standby” mode at least) and normal motor and electrical systems original to the DeLorean all work even without any plutonium in the chamber. We know this because Marty goes through a kind of check list before starting his final run:

Time circuits armed; flux capacitor... fluxing.

The reason why the car stalls so often (including dramatically soon after the above checklist) is partly for dramatic effect and partly because DeLoreans were not known for their reliability. And perhaps it’s meant to show how the time machine itself was cobbled together in a half-mad fashion by Doc Brown and maybe all the extra stuff has led to wiring issues and/or a greater strain on the whole electrical system.

The film's novelisation is based on an earlier version of the script and has a very slightly different version of events (Marty drives back to his then brand-new house and hides the car in the garage rather than hiding it behind a sign) but the broad principle is sound. The car is perfectly driveable with the plutonium depleted.

His eyes fell on the dashboard readouts once again. One in particular caught his eye. It was located directly below the Plutonium Chamber, a flashing light that blinked EMPTY over and over.

Shifting into gear and moving ahead, Marty realized that did not mean he was unable to move. It simply meant—

“Good God!” he said. “What does that mean? That I won’t be able to go back?”

As to why he didn't drive around in it, Marty is smart enough to recognise that his car is an eyecatching anachronism.

Recognizing the outline of a police cruiser, Marty quickly killed the lights and turned off the radio. It would not do, of course, for him to be picked up by the police. Even forgetting the fact that he had just arrived from a different time period, he would have enormous difficulty explaining the DeLorean, plus he did not have the necessary registration papers for it or a 1955 driver’s license. He wondered what the officers would say if he showed them his 1985 license!

...

“I’ve got it hidden,” Marty replied. “I stashed it in a garage. It’s so flashy-looking, I couldn’t drive it around the streets without getting a lot of attention. Maybe the cops would even arrest me.”

And he obviously does drive it again at the end of the film.

The other answers are too long.

• The car runs perfectly without plutonium (proof : At the end of the movie it runs all the way up until the lightning hits it). There is no indication of it running out of gas or anything. The temporary stalling is just for dramatic effect.
• The whole "Peabody shoots the car" scene is meant to explain to the viewer that this car immediately attracts attention (and trouble) in 1955.

Marty simply decides to keep a low profile, and hides the car.