First this hotel is NOT in Falls Church. It is in Merrifield, which is outside the Beltway. The map that displays on the Trip Advisor review page is flat out wrong -- put 8296 Glass Alley, **FAIRFAX**, VA 22031 into Google Maps instead to see where it is located.
The hotel is only a few months old, and is in an upscale new shopping district called the Merrifield Mosaic district. It has a number of good places to eat, and shop, but is otherwise an island of new development in an older commercial / residental / light industrial area. I've never been inside but have walked past it when shopping in the development, and it looks quite nice, but it is next to a major road and I'd wonder about traffic noise.
It is about a 15-20 minute walk from the Dunn Loring Metro on the Orange Line (Google says .8 miles), through a generally safe area. Late at night my guess is it will feel empty, but you are walking along major roads (Gallows Road to Lee Highway) so you should be OK. Once on the Metro you are looking at about a 30 minute ride to downtown; probably 50 to 60 minutes from Nats Park as you'll need to change trains at L'Enfant Plaza from the Orange Line to the Green Line. On Friday and Saturday nights the trains run until 2 AM, so you should be OK unless the game goes heavily into extra innings.
In terms of whether you should do this, you are going to be spending a great deal of time getting to and from your hotel, and not seeing the many things you want to see. I understand that budget restrictions may not give you many options, but if it were me I'd try to find a hotel closer to all the sites that fits my budget, even if it is older or has fewer amenities. What rate have you found for your nights? I'm happy to take a look and see what alternatives might exist in neighborhoods that are generally regarded as better options for tourists.