The dog you can buy near the Markarth stables can it die not by your hand? As in is it like other companions where it can only die if it's hit by you?

i would imagine its possible but extremely unlikely as enemies will completely ignore downed allies. i would recommend just using a mod that makes dogs ( and allies) immortal(but not invincible).

companions can die by other peoples hands, but people stop attacking them as soon as they go down and will only die to attacks from you or ones that missed/overshot their target and hit the companion as well (or they are dealt a VERY VERY VERY heavy amount of damage at once), and as far as I can tell dogs are the same.

I found Meeko and it died because of a trap in a dungeon

A giant killed my dog with its "you're going to fly club attack".....

....Since then im a giantslayer.