Welcome to bike commuting! Plenty of good comments on bikes, I'll try to address the route. The VDOT bicycling pages summarize the info on where you can/can't ride. Limited access roads are OUT, however 50mph roads (WITH a well-maintained shoulder, or slightly slower roads with a wide curb lane) are fine. Many people in NoVA use the Fairfax County Parkway /Rt 7700 shoulder, and I frankly think it's safer than the poorly-maintained side MUP, especially where it crosses 2 major side roads and the Dulles Toll Rd on/off ramps.

Looking at your map, how are the crossings to either side of Greenbrier Pkwy (the Battlefield Blvd or Indian River Rd overpasses)? Looking at the google maps overhead image, while there is a decent-looking shoulder on the overpass, the approaches and rest of the roadway appears to lack any shoulder of substance (typical VDOT design flaw, and it makes any backup due to accident or disabled vehicle MUCH worse). Are there any bus routes in your area you could use to cross I64? I looked into this when I thought I'd have to be commuting through an area where the most direct route crossed a major highway and is unbikeable. Alternates added several miles to an already longish ride, so I was seriously considering doing bike-bus-bike.

I'd suggest using tires known for puncture resistance as busy roads often have a lot of debris on the sides/shoulders. Regardless, carry replacement tubes (at least 2) and tools/pump so you can self-recover from 2 flats (in case you don't remove what caused it on the 1st tube!) and practice changing a tube using ONLY what you carry on the bike. And it sounds silly, but mount a decent size red reflector on the back (for VDOT-compliance) and consider running a strobe-style taillight (Planet Bike Superflash or clones) and bright clothes or vest for increased DAYTIME visibility. A mirror (helmet or sunglass mounted) is excellent for easily checking what's overtaking you, especially when riding near fast-moving traffic.