Europcar will allow you to drive to CZ, you just have to notify them (at the counter during checkin) and make sure the rental contract has a note that you intend to CZ. They basically jack up your liability to a higher rate or make you pay more insurance.

Your Indian friends will be checked at the border, but usually once they see the passport with the schengen visa you are free to go.

The checks at the border is mainly to detect so called 'schlepper' who are people who make it their business to transport immigrants/refugees etc across the border undetected for a fee. If you are a family of Caucasians they don't even stop you, if they think there is something they ask for passports, but you are waved thru after a quick check.