You can definitely go right over the drywall for the tongue and groove.
I'd recommend locating your joists and marking them out, then screwing furring strips into the joists, maybe every 16 or 24 inches on center, perpendicular to the joists.
This gives you a nice, flat surface to nail your planks to and helps with any minor drywall imperfections.
The old fence boards are a clever idea and can work, but you do need to be cautious.
First, make sure they aren't pressure-treated since that's a big no-no for indoor use. Even if they're untreated, they're used to being outside. Bringing them into a climate-controlled basement means they will shrink and move.
So to prevent excessive gapping, you'll need to let them acclimate in your basement for a couple of weeks before you start milling and installing them.
Using reclaimed wood is actually a riskier move than using new, kiln-dried planks, but if they're free and you're up for the extra work, I imagine it can be a really unique, rustic look!