They don't ask for any sort of proof and I think you will be fine. Do you have the tickets now or will you pick them up when you are there? If you are picking them up there, just go up to the window and get them without the boys nearby. Disney won't ask to see the kids.
Yes, you do need to make restaurant reservations and I do believe it's 180 days out. We usually eat breakfast in the hotel and then make lunch and dinner plans. You are better off making early sittings for both like a 11:00 lunch and a five o'clock dinner. Mainly because, most people are not trying to eat at these times. I would go online and check to see what the hours of the parks are. You probably want to follow the extended hours for each park and Disney rotates the operational hours for each park. For instance, go to Epcot when they close at midnight.
Are you staying on site? This decision will affect your dining choices too. The Animal Kingdom resort has some great restaurants and the Whispering Canyon restaurant at the Wilderness Lodge is great too. Do you want a character experience? These meals are usually more expensive but may be worth it if your kids are into the characters. Are you going for a birthday? Disney has special events for birthday meals.
Don't get scared by the guy saying you were behind. I called the first year we went and the woman was extremely helpful; asking what types of foods we liked and spent the time choosing the restaurants by the categories we liked. I believe I called the month before arriving.
Last year, I was on maternity bedrest but my husband took the two boys. He said that they all like T-Rex which is at Downtown Disney.
I would try to get into several of the Epcot restaurants, they are particularily good.
Good luck, have fun, remember this is your vacation!