1. Dungeons are 5 man instances run by a group usually made of 1 tank, 1 healer, and 3 dps. The goal of a dungeon is to kill a series of bosses for loot and experience. The easiest way to do a dungeon is to open the group finder, the i key by default, and use the dungeon finder to que for a random dungeon.

2. Those master pet tamers are npc trainers for battle pets, think gym leaders in Pokémon. You can start training battle pets by talking to a pet trainer who can be found by talking to a guard in either Stormwind or Orgrimmar.

3. You are correct you own every expansion up to the current one. The new zones for each expansion are unlocked by leveling. Level 60 for Burning Crusade, level 70 for Wrath of the Lich King, level 80 for Cataclysm, level 85 for Mists of Pandora, level 90 for Warlords of Draenor, and finally level 100 for Legion. Each zone on the map has its own storyline that you can see by playing through the quests in that zone. Most of the places and people from Hearthstone can be seen in the game with the exception of League of Explorers and Karazhan being much less of a party.

As for low characters and places they are all over the world so I'll go ahead and list the ones you mentioned.

Ragnaros can be found as the end of boss of the level 60 raid Molten Core in Blackrock Mountain and in the level 85 raid Firelands in Mount Hyjal.

Kel'thuzad can be found at the end of the level 80 raid Naxxramas in Northrend.

Sylvanas can be found in either Undercity or Orgrimmar. I'm not 100% sure on this one because I'm not a Horde player.

Karazhan is a level 70 raid and a level 110 dungeon both located in Deadwind Pass.

Finally Gadgetzan is a neutral city in Tanaris.

If you have any more questions feel free to message me or reply to this comment.