Hi S.,
I have 4 children and the 2 middle ones are boy 7 yrs and girl almost 4 years. When we go into a public restroom there are a couple things we do:
1) Always check the stall to make sure A) the toliet seat is clean B) the toliet has been flushed and C) there is toliet paper in the stall.
2) My son I don't really have to worry about because he stands up unless of course he has ro go #2 but then I don't allow him to A) open or close the stall door (we all go into the big handicap one, I have a 15 month old also), I do B) when he is done I don't allow them to flush the toliet (some places now have the automatic ones but some don't), I do with my foot and I don't touch anything else but the door to the stall and I give them toilet paper.
3) When we are done I keep hand sanitizer with me reason being, if I am by myself and I have 3 younger kids I can't put the 15 month old down she will take off and touch everything and the sinks are sometimes to high and I have to left my 4 yr old up. So I find hand sanitizer is better and I can squirt it on them when we get back to our table, to the cart or the stroller. If I don't have my youngest with me or dad is with me he keeps her while I take the younger 2, in that case I lift up my 4 yr old and get her the soap and turn on the water, I also turn on the water for my son. When they are done if we have to get paper towel I get them some if not and they have the hand dryers I turn them on for them with my shoulder or arm. After they have washed their hands I wash mine and then I do 2 things, if they have a hand dryer I once again use my arm or shoulder to turn it on or if they have paper towel I pull enough off to dry my hands but then either grab an extra paper towel to open the door with (if the trash can is to far away I just take it with me and try and find one somewhere else) or use the arm of my sleeve or coat to open the door.
And finally I have them go to the bathroom before we leave the house but even then they still seem to want to visit the bathroom like it is something interesting.