'It's rude to start to eat..' is not right. It's strange because 'eat' on this case is a long, continued action. So 'eating' is correct in this case.

It's rude to start eating before everyone else is (seated) at the table. The other one is OK and perfect, but 'to start eating' is more common.

That is o.k. too but it is more common to say ' start eating' rather than 'to start to eat'