Q: So I can't get any new gear if I don't raid in 5.4?
A: Not true! You in fact can get new 535 gear via the new Timeless Isle. You can get armor tokens from mobs that become iLevel 496 gear. This gear will be appropriate for your specialization when you use it to create the gear, so be in the spec you want gear for. However, you can also save the token and instead of making a 496 piece with it, you can gain a Burden of Eternity which will make a iLevel 535 item instead. So yes, by just soloing on the Timeless Isle, you can get your character gear that's equivalent to heroic Throne of Thunder gear.
Q: If Battlefield Barrens is going away, how can I get the pet/transmog gear/item?
A: Well, certain items will still be available. While the weekly quest and quartermaster are going away (and with them the latent Kor'kron armor items and resources like wood, oil, stone and meat) you can still get Gahz'rooki's Summoning Stone, the Raptorhide Boxing Gloves, and the Kor'kron Supply Satchel from the Kor'kron camps in the Barrens itself, which will not be going away in patch 5.4. However, this does mean that all other items sold by the Quartermaster like the Troll Shoes will be gone, and if you're working on the transmog sets using the Latent Kor'kron armor pieces this is your last week to finish. You also won't be able to get the Hordebreaker and Darkspear Revolutionary titles after this week.
Q: Can I still get the epic boot quest from Chen that current you get after the introductory Barrens quests?
A: We have been told that quest is staying in the game.
Q: How can I complete my legendary cape when no one wants to run Throne of Thunder/some other raid anymore?
A: The first 8 bosses in Siege of Orgrimmar will drop Titan Runestones now. They will also drop Sigils and Secrets of the Empire if you're not on the Runestone part of the quest yet. You can also run Throne of Thunder in LFR, since now ToT LFR bosses will drops the Sigils for the earlier stages of the legendary questline. So it'll be easier to catch up.
That covers some of the most often asked questions. If I missed one, the great thing about this post is, it has a comment section you can drop it in and I can try and come back and update.