I would probably use an inline fuse, and tap into the cigar lighter under the dash, rather than run all the way to the battery.
Regarding screwing into the dog house to mount it, that wouldn't be a problem provided that they don't protrude enough that someone can get injured on the screws when the dog house is off, or that they don't mess up any wiring or hoses, etc underneath.
I have had a CB in a number of cars, and I used to keep it under my seat with a cord that I would plug into the cigar lighter socket. When I want to use the CB, I'd throw it up on top of the dash, upside down. It seems, that they always mount the speaker on the bottom, and if the CB is not installed on the celing the speaker would be aimed away from my ears. The most likely place to mount it is on the bottom of the dash, so the speaker is amied at your feet. When throwing it up on the dash, it is closer to my ears, and the sound reflects off of the windshield. I have learned how to read the numbers upside down.
When it is not in use, it is hidden under the seat.
In my crown vic I have one of those all in the mic units. It stinks. It is nice and neat, but the speaker is so small, you have to crank it up, and /or hold it near your ear, or both.