Everyone, much thanks for all the strong information provided by each and everyone of you, thanks.

After going back and forth with wife, brother and mother in law, we have made the following decision:

We will fly from Basel airport to Rome on easyjet. Stay in Rome for 3 nights - 4 days. It will be be "easier" on the baby and mother in law, so we don't spend countless hours in and out of the car.

It was not my choice, I would rather drive around and get to know many cities, but, as said, Rome was non negotiable, and they were all fine with going "just" to Rome.

So we're driving from Zug to Basel Airport and leaving our car parked there, its relatively cheap (50-60chf) for the entire 4 days. (taking a train there would be very expensive for all of us...) and it is best to fly from there than ZRH because its a cheaper overall price (would be 900chf for us all from ZRH and will be only 505chf from basel), also I rather not fly an embraer, and last but not least, i like the times offered better too.. so win win in that scenario.

Now, I am off to researching the best transfer options from the airport to a hotel, and a hotel itself. Would be great if its very child oriented, and or close to a playground for kids and the vatican.. i guess I can find tons of info, and will start my research on it tonight...

Thanks again everyone for all the input. Didn't turn out to be how I wanted, but as most of you pointed out, It seems better/cheaper to do this way than my way...