On average, most American public restrooms/bathrooms/lavatories are not as clean or pleasant as equivalent public toilets found in Western Europe or Japan. Some may be pristine, such as in upscale shopping malls, fine restaurants, or commercial office buildings. Others will be shockingly unkempt, such as at many gas stations and bars. Some toilet stalls offer disposable toilet seat covers if you are bothered by this, but those are far from being universal. (The main reason is that most American public facilities do not hire a janitor who is tasked specifically with maintaining the restrooms full-time; rather, American restrooms tend to be maintained by a facility's general janitorial staff only a few times a day, with predictable results.)
Basically all public buildings are required to have handicapped accessible restrooms. Increasingly, restrooms offer baby changing stations in both the men's and women's restroom (these are mostly seen in shopping places and restaurants); a few places have a separate "family" restroom which is single-occupancy but spacious.
For little children who need to be monitored or assisted, it's generally acceptable for them to use the restroom of the parent they're with (little girls can go with dad to the men's room, and vice versa). The other way around (dad going to the ladies' room) is usually not okay.