1. いただきます is usually used before eating a meal.
So it's like. "Let's enjoy the food! / Let's eat! / Thank you for the food!"
たべましょ - used to invite someone to eat. ましょ- is used to make an invitation.
So, example. We are talking then you would want us to lunch. You would want to say:
ひるごはんたべましょ。Let's eat Lunch.
2. I think, both are used. いただきます － Said before meals. たべましょ － Invite someone to eat.
3. If you want to ask someone to eat, simply say.
たべたいですか。 Do you want to eat?
The root verb is たべる. Adding たい gives the expression of "want" so if it becomes たべたい it means "want to eat".