You can have one pet at a time, per party member. As long as the Charm Pet skill is in your skill bar, then it will follow you and show up when leaving towns. To remove a pet, it must be taken to an animal trainer. You can't get a new pet until removing the old one.
Yes, they level up with you. And sinking skill points into the Beast Mastery skill can have a dramatic impact on that pet's effectiveness, even if you don't take a lot of BM skills into combat with you.
I've had a strider a stalker and a wolf. There doesn't seem to be much difference. I like the strider the most followed by the wolf. But that's just my personal preference. The warthog can be tamed also but I haven't tamed one.
I think pets level on their own rate, independant of the player. Only because my arctic wolf has leveld 3x in the time it has taken my char to level once. The time it took me to go from level 12 to 13, my wolf went from level 7 to 10.