When you buy an animal companion, as a non-druid and a non-ranger, will it always be level 1? Or is it able to be increased somehow?
You cannot get an “Animal Companion” unless you have that class feature (from the Druid or Ranger class, or possibly from some other classes in supplements). You can buy, raise, and train animals, but they are not “Animal Companions” and gain none of the benefits of the Animal Companion class feature.
One of the benefits of the Animal Companion class feature is bonus HD; animals that are not Animal Companions do not usually get these, no. But they are creatures; you might check the animal’s Advancement section in the Monster Manual to see what it might get for experiencing combat. It won’t go very far, unfortunately.
If a non-Druid/Ranger wants something like an Animal Companion, the Wild Cohort feat is a pretty good replacement. It’s not quite as good as the Animal Companion, but it’s nice and general, and should at least allow the animal in question to survive the adventuring life, at least until fairly-high levels.