First if you intend to pray you must do it at the moment you start your prayer as else this isn't an intention (nyyah النية) as defined by some scholars, but could be regarded as a qasd القصد (an intention to do something in future) or 'azm العزم (a determined intention to do something in future). This means in our example you should make this intention in ...
The question according to Hanafi Fiqh is answered here. According to it:
The position of the Hanafi school is that the follower does not recite
any Qur’an behind the imam regardless of the prayer being prayed.
Practically this entails that when praying behind the imam:
One says the opening takbir (Allahu Akbar)
One says the thana (...
I am not sure I understand your question. You would do wudu the same way you normally do it. You would find a place to stop the car, take some water and do wudu. You may carry water bottles with you in the car, or you may look for a gas-station etc. In Muslim countries you can often easily find roadside mosques which would have the relevant facilities.