It really depends on your professions, since you said mounts are not a concern. So, I believe you would be looking at a few reps to help in terms of recipes, or 'helpful' items, and epic gear:
In MoP
- Tillers: you get your own food buff recipes and able to make Panderan Banquets
- Anglers: you get a raft and a mount that lets you run on water
- August Celestials: enchanting wrist recipes, epic gear for difficult slots (hands, bracers, also boots but see Shieldwall below)
- Order of Cloud Serpent: JC mount recipes (great at AH or for friends)
- Operation Shieldwall (Alliance) / Dominance Offensive (Horde): trinkets and 496 boots/belt
- Kirin-Tor: ring, back (both 496)
these provide epic gear but could probably manage without, and will probably replace quickly from running LFR and other reps
- Golden Lotus: chest, shoulder, ring
- Shado-Pan: head, back, trinket
- Klaxxi: legs, ring, belt, trinket (and 463 blue weapon at exalted)
And for myself personally, do the Argent Tournament grind and get the Argent Pony Bridle. It saves so much time from having to always get back and forth to a bank. This does not transfer across your other characters though, so you will need to grind it on all of them.
Other reps (especially earlier ones) are for epic gear or vanity items. Since, you no longer need to unlock AC or SP, you could avoid Golden Lotus. Shieldwall and Kirin-Tor (on Thunder Isle) will get you the higher gear faster.
Shado-Pan Assault is current 522 lvl, but that requires you to be doing Throne of Thunder raid, which you need 480 ilvl, so will probably need to get some nice drops or grind at least most MoP reps a little. You can get some decent rewards at just honored now.
UPDATE : with the recent patch 5.4, ilvl 496 is found A LOT and easily on the Timeless Isle, so you could easily forgo all but Shado Pan Assault gear factions above (you will still need to do Thunder Isle to advance the Black Prince rep/legendary line). Timeless Isle can be rough solo if under geared - many elite creatures with 5M plus hp and deadly abilities. Don't go as a fresh lvl 90 (if you even can) and try to go with a group.
Best Answer
Yes.
The loot of the weekly chest is based on the highest instance you finished, it does not matter wether or not it was inside the time limit (Source: I have tried it — multiple times in fact).
Blizzards preview is worded slightly misleading in that way.
Source: https://worldofwarcraft.com/en-us/news/20271497/preview-mythic-keystones
I am not sure if you get a chest if the only instance you ran that week was not in the time limit. but the above suggests that you need to complete any (even only a M+2) instance within the time limit for the chest to appear. But the item level is definitely based on the highest instance you finished, however long it took.