It's true! A Level 40 can battle in Pandaria!

[Dalandius]

You can battle in Pandaria at level one, but your likely to be level 2 by the time you get there due to discovery XP :
Only one of the 11 fights ( 7 Pandaren Tamers and 4 Pandaren Spirit Tamers) can not be done at at level 1 If you can get someone to fly you to the trainers, Seeker Zusshi is the only one you can not do.

One way to make use of this is make some alts on a server that is part of Cross-realm zones (or connected to) with yours. Get a friendly mage to port you the shrine and transport you to the trainers (Sand stone drake is a good mount for this), parking one at each tamer.
Yes you lose the charater Xp but I'm sure there are better way to level a character and you take, with an alt parked at each of the 10 fights you turn something that would take over 1 hour to something you could do in under 20 minutes.

To anwer your last point I do not think you can get out of the water near the Water Spirit even with a Water Skimmer mount, but leaving from Soggy's Gamble you can get to Pandaren Tamer Mo'ruk to the west. you may be able to get to Mo'ruk (you can get there from the Anglers Wharf but that has no flight point)

[Ranok]

Your post got me curious and I spent a fun hour with my level 42 Ally guy and a cooperative mage seeing if there were any other possibilities. Found two.

Can get to Mistfall Village and Aki the Chosen from the Shrine. Hug the Vale's southern wall. Did get aggro from a bird but was easy to outrun. At one point you ride through a battle in progress but they're all intent on each other and ignore you.

Also made it to Aki by jumping off wall at the Serpent's Spine flight point but got jumped a lot and was almost dead by time I got there. Run from Shrine much safer.

To anwer your last point I do not think you can get out of the water near the Water Spirit even with a Water Skimmer mount, but leaving from Soggy's Gamble you can get to Pandaren Tamer Mo'ruk to the west. you may be able to get to Mo'ruk (you can get there from the Anglers Wharf but that has no flight point)

Dalandius is right, it's impossible to get up on the rocks from the water and get to the Flowing Pandaren Spirit, pretty much need flying to get to him. But there's an obscure flight point way south in Krasarang Wilds, Marista, that's an easy run to Mo'ruk if you have a Skimmer or some sort of water-walking.

[Dalandius]

The Inn behind Aki the Chosen is a nice place to level by doing wild pet battles and is safe for even a level 1. There is 6 spawns and they respawn quite quickly.

Can also do the same with Farmer Nishi just have to be careful not to get the attention of either the wolf family nor the nearby virme. You can get surprising close to them before the aggro, there is also the plain between her and halfhill you can fight on.

[Ranok]

More experimenting with my Level 42 and it's a mage so is also squishy. He can handle two Master Tamers in Draenor once he got the necessary flight points.

Vesharr in Spires of Arak from the Veil Terokk FP. Can safely start battle with a Level 1 pet if backed up with maxxed Chrominius and Mech. Pandaren Dragonling.

Taralune in Talador from Retribution Point or Exarch's Refuge FP. Watch out for those damn moths, stay as far away from edge of the meadow making the run to the trainer.

Of course only way to do these trainers and the Pandaria ones regularly is a daily high-level mage assist. Otherwise no way to get to Stormshield if you hearth to Shrine, no way to Shrine from Stormshield.

[Kel]

The Inn behind Aki the Chosen is a nice place to level by doing wild pet battles and is safe for even a level 1. There is 6 spawns and they respawn quite quickly...

Been doing this for a long time now. Actually leveled 3 different toons from lvl2, 2 of them to lvl60, the other to lvl40. Even with full heirlooms the xp isn't worth doing wild battles past lvl60, so I bail on it at that point. But it cuts out all the early lvl stuff where you don't have flying. By the time you get to outlands you can fly so things go way faster.

I thought everybody knew about this?

[Paladance]

Reminds me about my lvl 60 DK hiding from the elementals in Twilight Highlands or the wasps in Uldum.

But that's because I'm too shy to call a mage…

[Ranok]

Reminds me about my lvl 60 DK hiding from the elementals in Twilight Highlands or the wasps in Uldum.

But that's because I'm too shy to call a mage…

Hey Dial-a-Mage anytime you need one. Usually too happy to help cause too squishy to solo.

[Paladance]

Hey Dial-a-Mage anytime you need one. Usually too happy to help cause too squishy to solo.

If only your magic would surpass the real life cross-continental limitations
But I'm sure that there are other people that need some help.