How should I run cables through interior wall to garage?

If you're within the city limits (or the donut) there are some NEC rules that will apply and you should probably consult a certified electrician as the NEC is vast and fairly dense. If you're just "in the area" of Bozeman but not within its jurisdiction, you're subject to rural code... which is no code at all. In this case use good common sense, spray foam any holes you punch down in to the crawl space to stop moisture from percolating up into the conditioned space, and if at all possible terminate your ends in junction boxes with an appropriate face plate so it looks professional and you should be fine. Also...cats drool, grizz rule.

It depends what you mean by "cables". As long as it is a data cord (not power) and the opening is fire sealed, it is fine. What I do is use a cutoff saw (hack saw is fine) to cut a length of PVC pipe. Use a hole saw to cut through the drywall. Then, caulk the PVC pipe to the dry wall. It makes a perfectly clean tunnel through the wall. You can feed data cables through this tunnel. There is no code violation for doing this as long as the cables are data cables and you fill the tunnel afterwards with fire proof sealant.

Note that you cannot have any "wall penetration" between a garage and a bedroom, but it appears from your diagram that the adjacent room is not a bedroom.