There is a curfew at DCA due to the airport's proximity to the city, so security may feel you have no reason to be there after hours (and may not even let you remain airside). Since I'm invariably dashing from work when I use it, I can't actually remember whether the seats have armrests or not.
There are quite a few hotels in Crystal City, the neighborhood adjacent to the airport (one stop on Metro, or a half hour walk if you are adventurous). While they generally aren't expensive by Washington standards, none pops into my mind as ideal MR lodging.
Washington Flyer discontinued its IAD-DCA shuttle years ago. The only direct way to go between the airports is by taxi, SuperShuttle, or rental car. However, you can take the Metro (Blue Line) to Rosslyn, then take the 5A Express Bus to Dulles. The 5A only runs once an hour, so you will need to plan carefully-- the Metro segment will be 10-15 minutes and the bus 45-60 minutes depending on time of day and traffic, but it'll be under $4 each way.
Alternatively, you can transfer at Rosslyn to the Orange Line to West Falls Church and take the half-hourly WashFly shuttle from there, which would be a little under $11 each way.