Asking University to reimburse renting a car
Yes, this is acceptable.
At many universities, adding a few days stay at your conference location in a far-away country (using your own accommodation arrangements) is allowed and sometimes even encouraged. Especially when this allows you to take cheaper flights. This would be something similar, using the car because it's required anyway, and adding a few extra recreational km you'd pay for yourself.
Renting a car might be an issue for some European universities, as they have rules to encourage public transport usage. The 1/4 share of the car is something you'll have to discuss with your administrative staff. They might want to reimburse 1 person for the whole thing, or split it over multiple projects. Just ask to clarify, they'll also be happy to know exactly how and what they're reimbursing.
Speak to your department in advance. While this is a very sensible thing to do, there may be rules to follow, ignoring which would add to the cost and/or complexity. These may include: Hire cars have to be booked through the university; the university's insurance will cover you so don't take it from the car hire company; or you must take the full insurance; or you must take the insurance abroad...
It's much easier for them to process one receipt once, so you may find it easier to use other methods to split the cost between projects (such as some projects paying for susbsistence). Again this will be easier if the budget-holder(s) are involved up front.
As a general rule, this is acceptable and common. All universities I have had experience with consider rental cars a reimbursable form of transport-related expense. But since we don't know about your specific university's policy, the sensible thing to do would be to ask them in advance of the trip to avoid unpleasant surprises. (E.g., at my university I once discovered after a trip in which I rented a car that I was supposed to use a specific rental car company my university has a special contract with; the department's staff then had to scramble to find some clunky workaround to help me get reimbursed.)
As for splitting the cost, I am sure your department's staff are capable of dividing numbers by four and other such simple arithmetic operations related to the fact that you are splitting the expense with other travel companions and are renting the car for a mixture of personal and work days. Just explain to them what you are trying to achieve and they will help you get reimbursed for the correct amount. Again, doing this before the trip would be good to ensure everything goes smoothly.
And as for the moral issue you mentioned... I have two comments: first of all, having fun while traveling to conferences is both moral and highly recommended. (Would you offer your university to pay part of the cost of the airfare because you enjoyed looking out the window during the plane ride? And have you considered why people bother to hold conferences in the middle of a hiking region?) Second, even on the days when you are only hiking with your friends/colleagues, I'm sure you'll be discussing your academic work, and who knows what will come out of it? Indeed, I've had many fruitful scientific discussions on hikes. So depending on who you're hiking with, even from the moral point of view one may legitimately consider this a form of work as well... ;-)