PATCH 11.2 AND SEASON 3 NOW LIVE - BANK CHANGES, NEW M+ DUNGEONS, ETC. - THE GUIDE IS NOW ONE YEAR OLD! (as of August 25th)
Here is a list of ALL dungeons currently available for lv10-14 twinks to enter freely (by either walking into or through the Timewalking dungeon queue). To access the dungeon queues for Chromie Time Classic dungeons, you need to enter the Cataclysm timeline. There are no Timewalking events beyond Battle for Azeroth TW.
NOTE: Battle for Azeroth Timewalking is very new, and while there was some testing in PTR, there are bound to be bugs with unscaled damage mechanics and the inability to interact with certain chests and mechanics. Please report these issues when you find them, as they will affect anyone under level 50.
As a reminder, you can only get the suggested socketed gear (with the exception of rare BOEs) from the Normal version of the dungeon. However, as detailed on the other pages, there is gear from Timewalking dungeons that is also exceptional. Again, unless specified as being from Timewalking, assume suggested pieces are from normal dungeons.
There are technically two more dungeons 10s can play in the form of Culling of Stratholme and Halls of Reflection, but this is through Lorewalking. This is a list of dungeons you can enter through unexceptional means.
For mounts obtainable in other ways on low level twinks, check out Conzil's F2P Mount Collection Guide. This list extends to level 20s as well.
If you managed to unlock the Campaign skip, and want to possibly lock yourself into receiving the bugged ilvl 252 loot from SL leveling dungeons, you can do Torghast when you do the campaign skip on a level 10. This is ill-advised unless you have no expectations to farm loot in Plaguefall, The Necrotic Wake, Mists of Tirna Scithe, or Halls of Atonement.
Soul Ash (it is not possible to use Soul Ash to craft legendaries on level 10s).
Soulshapes (if part of Night Fae covenant).
Tower Knowledge for the Torghast skill tree.
Buyable Cosmetics (req. lv60)
Any Jailer's Gauntlet rewards (can't enter?)
The higher floors with harder content.
Access to the Runecarver.
Access to the Maw via the back door.