Happy to see a new-comer to the series! I hope you'll come to love it just as much as the rest of us here
Now let me see if I can help you out with answering some of those questions for you!
So, essentially in Vanilla (modless) Skyrim, you can have 1 follower and 1 pet, like our lovely friend
@Hart already said. Followers are any humanoid NPC in the game--man, mer (elf), and beast-folk alike. Not everyone can be a follower however, but there are quite a few who can be. Gaining a follower can be as simple as finding a hireling/mercenary like Jenessa and paying her to be in your service, making a friend by completing a quest to help them like Mjol the Lioness, or simply being given one like in the case of the Housecarls. Best way to find out who can and can't be is through just playing through the game and fluidly finding out, but if you are like me and would like to do some of your own research, I'll provide a link below for you to look into!
Pets are more limited in the game, but essentially follow the same ideas of the followers. You assume right by guessing dogs etc. They are vast in options, and can be gained through the same means of the followers; either by purchasing them, helping them or being given them. Pets are a little more difficult to find however, so there'll be some info in the link I provide for them as well.
Now, without any mods, you can't have more than 1 of each - so in your case with the follower mod you have on it seems you are able to have her and one other; which is pretty cool if you ask me. But if you want to have any more than just that there are plenty of mods out there that will allow you to have more. However, unlike in DA, the followers are a little more cumbersome in my opinion, and lack the same versatility that they would in Inquisition for example (unless again, you have mods installed to change that).
So when you are adding a new follower, or rather switching them out, you do in fact have to swap them out. That means that with paid followers, they will leave your service and go back to where they first hired them--and if you wish to get there help again you'd have to pay them again. With "friend" npc's who you helped with a mission, or gifted npc's like housecarls, when you replace them they will either go back to the city you found them in, or sometimes even whatever player home you may own (not including faction housing). If you switch out a follower that doesn't mean that they can never be a follower again though. Getting them back is as simple as talking to them and asking them to follow you again!
Hopefully this answers some of the questions you have, including the ones you asked. As more if you need, this forum is your recourse friend
Here's that link I promised:
http://en.uesp.net/wiki/Skyrim:Followers