Obviously when travelling by public transport, all I need to cross non-EU Balkan borders (meaning Albania, Bosnia, Kosovo, Macedonia, Montenegro and Serbia) is my identity card.
However, what if I were to rent a car in Switzerland and wish to drive to these countries? What exactly are the "basic" things to show at passport control besides the ID?
From what I know, it's just the rental contract. If that's the case, is it always asked for along with the ID, or only occasionally?
Your first problem is not at the border, but to find a car rental company in Switzerland allowing you to drive their rental car to any Balkan country. Different companies operate with different lists of allowed countries and in some cases, the list of allowed countries also depend on the vehicle class, but the Balkan area is an absolute no-go zone for most rental cars.
If you against all my bets should manage to find a rental car, which you are allowed to drive to the Balkan countries, you must be prepared to present the following documents at the border:
Car registration certificate.
To enter Albania, Bosnia, Macedonia or Montenegro, you need proof of 3rd party liability insurance in form of a CoBx Green Card.
To enter Kosovo, you will have to buy additional 3rd party liability insurance at the border.
At least for Kosovo (but this may also apply to other Balkan countries), you need if it is not your own vehicle, a notarized declaration from the vehicle owner that you are allowed to take the vehicle to country x. A rental contract clearly stating that you are allowed to drive the car to Kosovo may fulfil this requirement, but I would not be 100% sure.
Except when entering Kosovo, the paperwork is only occasionally checked at the border.
In addition to the rental contract, you need the "green card," the international insurance document. You must tell the rental agency which countries you intend to take the car into to ensure that you are allowed to do so and that the car will be insured in those countries.
In some countries, at least, you are unlikely to be asked for these documents when you cross the border, but it can happen. I've made a few dozen border crossings, if which maybe half a dozen were in rented cars, and I've only been asked for the car's documents once. However, in almost every case the car had been rented in one of the Balkan countries, which isn't your plan obviously.
The countries I've had experience with are Bosnia, Croatia, and Slovenia, both before and after the latter two countries joined the EU. The cars have been rented variously in Italy, Croatia, and Bosnia.
