I think you'd be better off trying to see fewer places at a slower pace. I spent all afternoon at the Temple of Heaven complex in July. I think we walked to Tianenmen (sorry I'm too lazy to check the spelling) and then took the subway back to the hotels. We spent about two hours in the Temple of Heaven. It could be done faster if you just went to the main three buildings. There were long lines to see inside the two main buildings.

I'd rather have time to sit in the parks and rest, particularly with small children.

I don't recall having a problem finding toilets. There were several in the Temple of Heaven complex.

I didn't go to the Forbidden City.

I don't remember any particular security in the subways. They were checking bags, but we just walked by.