Yes, if your server feels dead, it is worth re-rolling to a high pop server as you are doing. It will make a huge difference in your options and experience at the level cap.
No, just let that character go. Unfortunately "new player" servers are like this, because of the way Blizzard uses that tag. As long as they are tagged, they boom in the short term. When it is removed, they crash as many new player lose interest, quit the game, or migrate to more viable realms. Then the tag gets reapplied when the server is dead again . . . lather, rinse, repeat. They almost never manage to develop a steady, high population of players that stick around long term.
You're doing fine. It will take some time. Brewmaster will make the queues go much faster. A variant on this that you might find less tedious if you hate it is to only do each dungeon once when it is yellow in dungeon finder (in your level range), and then just quest at other times until new dungeons are available. Much of the XP available from leveling dungeons comes from the one-time bonus quests, so if you only do them once you will get a lot more XP per hour for the time spent in dungeons. This also makes the whole thing seem like less of a horrible grind.
If you like playing the game so far, then I believe you will be much happier with a new character on a more active realm. Windwalker is quite good now and fun if you want to DPS. Brewmaster is a very good tank, though it had a rough patch at the start of Legion. Some would say it takes the most skill to play well, but in the right hands also can be one of the strongest tanks in the game, and the most fun to play.
The best tank is the one you like to play the most, because that is the one you will play best. They're all viable. Guardian may be the most forgiving for someone new to tanking. Veng recently was buffed and is very strong now. Brew I already mentioned. Warrior you already know; it's the most traditional sword and board style tank, and probably the simplest to play aside from Bear. Paladin also is very standard, with a few more tricks that can be used to cheese some mechanics. Blood also is good, though it has less straight mitigation and relies more on healing back damage; it's excellent for farm content, but can be demanding in progression.