Do I have to pay traffic fines issued by foreign governments?
Yes, that's the law.
Next year I'm planning a long stay in the US (around 6 months) and I want to bring my car with me.
This is a whole other question, so I'll take it instead as a factor in your main question rather than go into the pros and cons of shipping a private vehicle.
If your aim is to avoid the consequences of breaking the law then breaking it in a vehicle easily traceable back to you will make it harder to dodge.
I was wondering, can I abuse carpool lanes, not pay parking spaces, break the speed limit etc etc with no consequences (while still driving safely ofc)?
There are prescribed consequences for breaking each of these regulations and those will definitely vary from state to state and quite possibly even from city to city. Some places are known to fiercely apply these rules or have periodic crackdowns. You will surely be in more trouble for things you are pulled over for than for things you find a ticket on your windshield for.
In any case this is a vast question about which entire books could be written, which makes it not a good Stack Exchange question.
If they give me a fine, is it just trash paper or can they somehow force me to pay?
I'd say it's a legal document but I'm not a lawyer. If you decide to treat it as trash paper you cannot expect us to condone such an action. Whether they can force you to pay surely will depend on many factors and is again too broad to try to answer here.
(What I want to avoid is basically going back to the US 20 years from now and getting arrested for that, or something similar)
To avoid future negative consequences 1. Don't break such laws or 2. Be accountable if you do break the law and pay the fine.