As for your question about what to bring, ask the host if you may bring fruit for dessert or something like that. That seems to work well and if you bring several selections, or a fruit salad, there is sure to be something everyone can have. You may inquire about food alergies when asking this question, but we have never had a problem with this tactic. Sometimes they will tell you not to bring anything, and this is okay too. You might put a game all children can play in the car with you, or offer to bring it.
I always offer to help with clean up when we go somewhere. I am usually turned down, but at least the host knows we are willing to do dishes, etc. I do ask my child to put up or put back what he has been playing with before leaving if that seems to be the norm (if there is a place it came from), such as a trunk, closet, etc.
Also, reinforce whatever rules the host sets. If children are to take thier dishes from the table to the sink, then have your child do that. The worst thing is when a parent lets thier child do something different than what you've set as the expectation in your home.
As for the dad question, we tend to do whole family get togethers in our circle and dads are invited. Sometimes they don't come, but that seems to be the exception rather than the rule.
Remember to send a thank you note right away, and most people seem to do that by email now.