Generally
Logos, names and body designs of cars are trademarked and can't be used outside of situations explicitly stated in trademark law (while this may slightly differ per country it will in general mean - no, you can't use them in your game, even when your game is being free).
The only way to use them is to contact the owner of each brand and ask for permission. You will most likely be presented with an agreement, which will contain conditions under which you can use each brand. Each of the brands owners will see this as an advertisement and will have specific conditions for each of their cars (whether the game is free of charge or being sold doesn't matter - it is still an advertisement).
Just to describe an example - in Forza 4 team there was one full time employed person whose responsibility was to make sure the cars in the game looked like and matched the performance to various licensing agreements they had with multiple companies.
My experience
I have been in similar situation in the past with non-game project. It was an advertisement project for single car company, now while it is clear we didn't have to pay any fees (as the project was made directly for them), we still had an agreement which included quite long list of requirements for quality. They had a person which was regularly checking our compliance with all requirements.
Due to legal agreements I can't tell you exact details, nor I can show you how such list of requirements look like (it will be longer than you expect) - but in general it ranges from specific things like allowed body colors, through specific features that you must show (which are often main selling points) to having certain level of quality of how the end result looks like.
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Do NOT try the way where you use their trademarks without written permission even in free game, stating that no harm was done as the game is free. Court will dismiss such argument. You will not win the lawsuit.
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In short: You must consult owners of trademarks. If you're presented with agreement or offer, I recommend you take a lawyer with you. If you agree on using trademarks - you will have to satisfy their conditions.
