As a 10-year suburban bike commuter (plus 8000+ miles of cross-country bike touring, including through Charleston a year ago!), I'd say that on the whole, this actually looks like a pretty sweet route. Starting from your house, going to work:
- Foster Creek Rd., might be one of the worst sections, since there are no shoulders, and the lane isn't super-wide, but it should be wide enough and the intermittent median gives extra room for cars to slide over.
- Tanner Ford Blvd. looks pretty good, Street View shows there's plenty of room for a car and bike side-by-side. Though with the concrete curbs (unlike on Foster Creek), make sure you don't get squeezed *too* close to them.
- Rhett is pretty much an ideal cycling road. Big-ass shoulders (including over the bridge), so you should have this awesome "bike lane" all to yourself, and if there's lots of fast-moving traffic, you can even get a sweet draft off of that to make your pedaling easier.
- Once you cross the bridge, the marked shoulder unfortunately disappears, but it looks like it maintains a wide outside lane for the whole stretch, which is nearly as good.
- Remount Rd. looks the worst, but it's a pretty small stretch of the whole ride, so should be survivable. I generally use left turn lanes like cars do, and despise sidewalks, but in this case, depending on how tight the lanes actually are and how much traffic there is, I might do the "pedestrian left" from Rhett to Remount, and then take the sidewalk on the south side of Remount. There aren't *too* many curb cuts along that section of sidewalk, but there are enough where the chance of getting hit by a car entering/exiting one of those driveways is still likely greater than the chance of a car rear-ending/sideswiping you if you just stayed on the road.
- Then on the way back home it would be a decision between crossing back over to the south side and taking the sidewalk back (even *more* dangerous at the curb cuts because drivers exiting the driveways never look for a cyclist coming from the right), riding in the grass on the north side, or just riding in the road. That's the sort of thing where experience will be your guide, and you'll weigh pain-in-the-ass-ishness vs. perceived danger. In my experience, you get less scared over time about being in the road, and eventually you ditch the sidewalk/grass routes for the more-efficient direct routes.
The #1 key for this route will be a mirror, particularly a helmet-mounted mirror. Most of the time it will just give you confidence by making realize that all the cars coming up from behind you *do* see you and give you space, but then it will also be critical if you want to find gaps and slide across into left-turn lanes.
Edit: as far as *when* to ride, do it whenever Google took its Street View photos and aerial photos! There is essentially NO traffic in any of their images, it makes me wonder if you're making all this Death Truck stuff up. :-)