I recently learned that to express refusal people say won't/wouldn't.
For example, My car suddenly breaks down and I say "my car won't work".

My question is:
What would a native think if they hear I say "my car doesn't work"?
Would I be understood?

Question

I recently learned that to express refusal people say won't/wouldn't.
For example, My car suddenly breaks down and I say "my car won't work".

My question is: 
What would a native think if they hear I say "my car doesn't work"?
Would I be understood?

Answer

Hi. ‘Won’t’ and ‘doesn’t’ could have a slightly different meaning, related to the reason. Example:

My car won’t start. (I tried to start it, I asked it to start. It refused to start. I might be tempted to say this if I’m not sure why or I don’t care why it didn’t start)
My car doesn’t start. (It’s just a fact. I could very well know why it doesn’t)

I would never say ‘My car won’t work’. to mean that ‘my car doesn’t work’. It sounds like I’m making a prediction about something in the future. For example:

A:Can we use your car to get to the airport?
B: My car won’t work. It only holds 5 people and we have 6. (It’s not suitable for that future activity.)

If you want to make a current factual statement about your car, you could say:
My car doesn’t work. (It won’t start./It doesn’t start/The battery’s dead/I blew a gasket/It was in an accident)

Answer

"My car won't work"-- An expectation:  I think it won't work, but I haven't tried starting it yet and am not necessarily  sure of the outcome
"My car doesn't work" -- A fact. The car currently isn't working and possibly hasn't worked for some time
"My car isn't working"-- A current state of being, suggests that this development is recent

Note:  I have also heard people say something like, "My stupid car won't work."  In that context,  "won't" is used to mean "isn't", ie, "I've already tried starting the car and it won't work."  
However that's a colloquial, spoken-only usage. I wouldn't write it that way unless I was purposefully writing dialogue or the narration of a person and wanted to indicate that they were speaking in a casual or slightly ungrammatical way.

Answer

Probably what you're picking up on is a regional thing, where people from some countries might prefer to express the same thing using slightly different words. I think in Australia, where I am, we'd probably say "my car's not working" or "my car doesn't work."

Answer

My car isn’t working is correct and probably more common.

You hear « my car won’t start » often. That means the engine won’t turn over. Maybe no battery or out of gas.

Answer

Yes, we say that.
You will be understood. 
I would assume that your care needs some kind of repair.

Good luck!

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