Hi HLW! Nice to have you here on the Dragonvale forum!
I'll answer your questions in order. Also hope some of our forum friends can help me out on a few of these if I am mistaken/confusing:
1) I think the number of visitors just shows how popular your park is. You can make it better with more dragons and decorations. Then again, I never really pay attention to it much. It doesn't do much besides that.
2) You are correct. As they level up, they change their appearances and get bigger until the dragons are level 7 (adult form). Once you unlock the Colosseum, you can put a dragon in there to get exp, gold and gems once a day. The higher the level, the more likely that you will get 5 gems, a lot of gold and a lot of exp for yourself. And then when you unlock the element shrines (starting at level 18), your dragons can go beyond level 10 (the max before then) after you get 50 dragons of that element to level 10.
3) Yes, you are also correct. Treat farms can also be upgraded when you are at level 10 (large treat farm) and level 16 (huge treat farm). There will be more treats available for you to grow.
4) Upgrading the nursery will give you one more space to put your eggs that you get from breeding. It is recommended that you do this as breeding fails will start happening soon lol! And don't worry about using up all of your gems, we can all exchange gems through the gifting option. Just send us friend request by putting our Game Center ID in your game center account. Once we accept it, you will be able see our parks and be able to gift gems in the Friends button (bottom right hand side) in your Dragonvale game. Oh, and you can get gold from our parks (symbolized by 3 party hats).
5) I would not be surprised if they did! Dragons are magical creatures after all.
It's a lot to take in, but this will make sense (especially #2) as you level up your park. And you can always ask us for help here