I rode all winter last year, from Fairfax/Vienna to DC, using a variety of routes and modes, including the W&OD, the Custis Trail, the Capital Crescent Trail, and the Tow Path. As far as I can tell, the trails are not cleared, though I think Falls Church City might have done something, as that section seemed to be better, overall (but it could just have been sunlight). Conditions are hit or miss, though, as DC winter is quite variable. If it warms up quickly after a storm, you can get substantial melting in short order. OTOH, last winter the ice seemed to linger for weeks. The W&OD is a rail trail, so the hills are pretty gentle. Less so on Custis and perhaps 4 Mile (I have less experience there).

I rode all winter on 700x37c Continental Country Rides, and wasn't smart enough to reduce pressure. I took one very low speed spill, had a couple of close calls, spent some time with one or both feet steadying as I slid down hills, and generally just soldiered on. I did start using alternate transportation more often the day or two after a storm, until I knew we had some warmer weather, though - it was NOT pleasant.

You don't specify where in 'tysons' you'll be working -- there a lots of places that folks speak of as 'tysons' -- for this purpose, I'm assuming you work somewhere very close to the intersection of rtes. 123 and 7 within close proximity to Tysons Corner Center, or the Galleria (International Drive, Westpark Drive, etc...). Tysons Traffic can be quite daunting, and would not be my first choice. I considered taking a job there (off international) and was glad I had another option, as I wasn't sure how I'd be able to keep riding to work safely. I was planning to give it a go, though, and I do NOT think it is impossible.

I've ridden parts of gallows towards tysons, and honestly, it isn't that bad -- routes 7 and 123 are much worse. I saw a guy biking on 123 past international drive the other evening, and I have to admit I thought he was crazy. As others have mentioned, riding in downtown DC is MUCH more pleasant, as is most of Arlington.

On Gallows, in the morning, I would not be reluctant to ride the street there, and you have the option of the trail along the side. I would be concerned, however, about crossing each one of the many entry/exit points -- I find this a MUCH larger hazard than the mostly straight moving traffic. I would expect to hop up and down, street to side trail pretty regularly, as I became accustomed to the specific hazards of the route.

If you were actually going to Tysons Corner center, or one of the surrounding offices, or going to 'tysons 2' area, then I'd recommend turning on Gallows Branch to Kidwell/Towers Crescent, which heads to 'Ring Road' into the mall; Ring Road has a stupid little stub of a bike lane but it doesn't go around much of the mall, However, it is a slower moving area, and it's a relatively easy way to cross rt 7 and 123 via Westpark Drive.

Now, having said all this, I'm not as sure this would be good in the evening going home - one advantage in the morning is that you don't have shopping traffic. But in the evening you have the double whammy of folks commuting home and shoppers/diners heading to the tysons area as a destination. I would second the folks who advocate for maybe finding a bus from your workplace to a point near the W&OD trailhead at Gallows.