What time are you arriving? That might make a difference.
From Domodedovo there is an express train which most people are entitled to take for free with their airplane boarding pass. If so this would make your travels much easier as from the Paveltskaya train station it is an easy transfer to the Paveletskaya metro station and just a few stops to the center of town and the Tverskaya street metro stop. Check the airport website, DME for information on when the express trains operate.
I have been to Vnukovo but not in many years (1995?) so I can't tell you anything about it since it has become an international airport. DME, however, is modern.
The Red Square hostel is not in Red Square but it is on Tverskaya Street near the metro stop of the same name.
Ruth