There is this ugly mutt who follows me everywhere and is handy in a fight.

My question is: If I send her home where is that?

Is she gone unless I re-find her in the boondocks where she first turned up? She is presently waiting with Lydia in Breezehome while I do some stealth stuff. I don't want to lose her because I love that devoted look in her eyes when she looks up at me.

Which dog is it?Where did you find him?

In the mountains across the river (East) from Karthwasten. She is just called "dog", IIRC.

She started tagging along when I helped her fight a pack of four wolves.

I left "dog" at Breezehome, told her to 'wait' while I went out to do one stealth job. Back at Breezehome "dog" was still there. I quit Skyrim, ending my day's session. Resuming today, "dog" had gone, apparently gone back home, wherever home is. It looks like dogs as followers are very temporary indeed.

The unnamed dogs you randomly encounter really don't have a home to go to, there are 2 named dogs that do have a place to call home- Meeko and....I forgot what the dog near Markrath was called. But Meeko is free and you find him in a location called Meeko's Shack and will follow you where ever for free, the other is a war hound that you need to buy for 500 gold, and if you send it home, you'll have to buy it again for another 500 gold.