I would suggest for starters, keeping the antenna location simple and
the cable run short. You may not need the extra height from the TV
antenna tower. Do be sure to keep the chimney mounted antenna clear from
the soot that comes out the chimney.

You did not state the construction of your house, but regardless, I'd
say go ahead and drill a hole through the wall.... perhaps down low near
the base boards like the telco does. I stuck a piece of 1" PVC through
the hole I drilled and put PVC flanges on each end to hold in place in
the wall. Seal with tin foil, steel wool, etc. to keep small critters
and bugs out.

For cable, I'd suggest RG8 (foam dielectric preferrably) or RG8X
(smaller). Either should be fine for 2M and 440 operation. For a 20
foot run, you could even get away with RG-58. Its lossy, but probably
won't be an issue on that short a run.

For a really cheap, but effective antenna, I also like to use ground
plane made from brass welding rod, on a SO-239 coax connector. Standard
PL-259 connector fastens right to it.

Nothing wrong with using a battery for that rig..... perhaps you
have a small 12V power supply.... 1A rating or so... would keep the
battery charged. Some larger 12V DC wall warts might suffice, too. Be
sure you have adequate venting when using a lead acid battery though.

Good luck.

Ed K7AAT