Hi. I'm not really new, but I made an account and haven't been super active. I've been wondering one: How do you get silver coins and two: Can you only get pets through breeding, catching, and adopting through the pound/trading?

you can get coins through exploring and quests. and for every time you post you get ten silver. there is only one other way to get a pet by holding the portal on the battle feild. other than the ones you listed

You get some silver every day when you log in that comes from your townhall (more, if you build higher level townhall) + if you have an inn and a guest in it, you get 50s/day.

The most common and easiest way of getting more is from explore, by completing quests and trading (pets, items). There's 1500s/day to get from voting for BK page. You also get 500s for every pet sent to pound. During some events, like the current one, you can exchange event tokens for silver (altough I wouldn't recommend it).
Less common ones are betting silver on your pet in the arena (never tried, so I don't really know how this works :P). Also, you get a few tousand by holding certain points on the batllefield for 10 hours.
I think you also get coins if you create custom skin and people use it for their pets.

As for aquiring pets - it's what you wrote + as saberia12 mentioned, you get one wild throdama or symurgh on the battlefield, for holding the portal for 10 hours. And technically, there are also Kami and wild lykos, but that's only from events :P.

Look through the player guide, in the forums, for your silver question, though they've mostly been answered. -- The one you'll want to read is: Guide: Making Money

As for getting pets, there are a number of ways; Buying/trading, the three you mentioned, and earning -- Which like the other two stated, can be done a number of ways: Battlefield Portal, events, or winning in Exhibitions.