Let's examine them in order
Josh wouldn't stop being a gentleman for money.
This means that Josh wouldn't accept money to do something that a gentleman wouldn't do. For example, if Josh were at a bar with his friends, and they promised to pay him if we said something inappropriate to a lady, he wouldn't do it. This use of "would" here is to express a hypothetical - you could rephrase the above as "Josh wouldn't stop being a gentleman even if he were offered money."
Let's look at the next one:
Josh won't stop being a gentleman for money.
Here, the use of "won't" is to express the speaker's certainty about the future. Maybe Josh is about to actually be offered some money to do something un-gentlemanlike. But the speaker knows Josh well, and is sure that he won't take the money.
Josh doesn't stop being a gentleman for money. (notice that don't would be incorrect here).
This is a statement about who Josh is as a person, based on the speaker's experience with Josh. Perhaps there have been situations where Josh was offered money to be less than a gentleman, but it never worked. So now it is an established fact that money doesn't affect Josh being a gentleman.