Do you miss distress calls if you let your partner drive?
Letting your partner drive has obvious advantages (shorter rides, no chance to do damage). But are there any negative aspects to it?
Do you for example still get the distress calls during missions?
Yes, you "miss" distress calls if your partner drives as this is pretty much a "fast travel" option. You will also miss out on finding landmarks of importance.
I have no idea if this is made up for later on in the game if you decide to ignore all of them on a case by case level (as in if you choose to miss 1 distress call does it get moved to the next case?).
Yes, you still get distress calls during missions. Think of it this way, when you are on a case, no matter what part of that case you are up to, you are still in the mission. When I last played the cases seemed to be "spaced out" in the sense that you could only get 1-2 distress calls per case. It could have been my luck though!