Huh, that's a good point - a large enough RTG on a data uplink cable would make its own way down through the ice, given time. There would be a lot of crosscontamination between the various ice layers and the lake though, so that's unlikely to fly (or sink, as the case may be).
Not necessarily. The hole will be stationary relative to Mars so you could just send a little base station with radio and power that drops a paper towel roll sized module with an unspooling data/power link cable and your science stuff into the ice. It can either bore its way through like a TBM (hard but good scientific value since you can measure environment on the way down) or just have it heat itself and let gravity drop it through (no moving parts but less scientific value since you're altering the environment around the probe). When you hit water stop unspooling the cable and just hang it there by the cable.
I'm sure college students in Maine or Montana would be able to pool their ice fishing expertise and come up with some working prototypes if given a sufficient quantity of beer.
It's fun to play armchair engineer.