For those of you unfamiliar with the southern end of the MVT:
At the crest of a rather longish hill you will come upon a few benches, a water fountain (which probably gets shut off during cold weather to prevent freezing), and a few, somewhat useless racks. If you continue along the trail however, you will find a lot more useable racks at the second entrance to the parking lot on the left where the trail ends. You may also continue on the sidewalk towards the visitor center and park at the rather small, old school style bike rack that is next to the visitor center on the left hand side.
Inside the visitor center are public bathrooms and water fountains. There is also indoor/outdoor seating in the food court as well as a few tables/chairs/benches outside the visitor center. The Mt. Vernon estate is open 365 days a year as is the visitor center.
You can’t enter the estate itself without purchasing a ticket.
Next to the visitor center is a much more upscale restaurant. Given that they advertise candlelight dinners I’d probably call ahead to see if cycling clothes are appropriate there.
If you need bathroom facilities before you reach the end of the MVT, the pavillion at Fort Hunt park just a couple of blocks off the MVT is a good place to stop. They have multiple stalls, lighting, and regular wash basins. The facilities at Riverside Park further south leave lots to be desired. Fort Hunt is also a good place to go if you get caught in a thunderstorm.
I also recommend walking (don’t ride!) the Dyke Marsh trail near the Belle Haven marina. It has a nice boardwalk at the end.