The underside of the terminal track looks like this:

Removing that cover exposes all of the electronic goodies that make your Sport track work, along with the offending wires:

It's best to not mess with that circuit board, so I cut off a portion of the cover and reattached it:

Then it's merely a matter of heating up the solder on all but the black wires (the black wire doesn't cross a slot, so leave it be) and yanking the wires off of the rails.
Taking the other half of the cover, I lined it up where the wires needed to be, then drilled holes for them. I also cut a notch for the wires to pop out of the terminal track and head behind the cover.
After snaking the wires through the freshly drilled holes and resoldering, here is the result:

It was at this point that I realized that I should have cut the cover here:

...instead of where I had. The terminal is flopping around now, and if I had cut it at the above line, I would have been able to attach it to the track with the first two screws. Oh well. Learn from my mistake.
The end result was still a positive as there are no more wires to foul up slot car guides. All in all, this is a success!

The fix for the lap counter is actually a lot more simple, without the need for a soldering iron. First off, remove the cover to expose the wires. Be sure to keep the track piece flat as the flippers that are used to trigger the counter can fall out.

The first thing that I noticed was that the wires were a lot smaller than those on the terminal track. I didn't want to mess around with soldering these. But, the wires themselves were melted in to place on the underside of the track:

A little finagling with some tweezers, and they all popped out. It is only necessary to unearth the wires leading through the Lane 1 slot. Leave the other side of the track alone.
Having learned my lesson with the terminal track, I cut the cover off just after the first set of screws that connect the counter to the track:

Since I already decided that I didn't want to play around with soldering wires this small and threading them through new drill holes on the cover, I merely used the dremmel to hack out a couple of notches. This was all it took to ensure that the wires did not interfere with the lane 1 slot:

Ta-dah! No more wires in the slot:

And with that, I'm off to finally run a Slot.it car on my track without hearing a "clunk" and watching it go tumbling. :D