Is there a way to maintain your MP and TP during battle? Most mages have a skill at some point that helps them refresh their mp even by a little and... though there isn't always one in parties, bards can also help regain mp back. As for TP I'd suggest eventually trying to cross over Lancer's Invigorate skill which is gained at lvl 22 to archer/bard which recovers 300 tp and cooldowns for 180s, aside from that I don't really know of any tp pots or anything. @-@
How do you get a house? What are the costs and also what are the benefits to owning a house? Not much is known about this but apparently there will be gardening, player selling and decorating going on in the housing area, I heard there will also be personal and fc housing but the price is currently unknown. (Im also pretty excited for it!) 
What's the quickest way to level up? As some people said, hunters log would be a very good thing to catch up on and some fate spots are more preferred than others I've heard so... I would suggest doing Aleport fate partying which is in limsa if you can get there, at 15 I think players are given airship pass so... its basically located in limsa. ^^ (Overall the best way is the storyline for a player whose just starting out though.) Oh and just a warning the story will jump in levels around... 38ish so fating is recommended at that point.
When should I start learning to craft etc.?
Crafting is pretty good to catch up on though its kinda hard near the 30s / 40s of it if you go pure crafting with no leves done... so I'd suggest doing it with leves if you choose to. I don't think theres a level for which... one can or can't craft though it can break a pocket but it also makes money if you know how to use the marketplace well, It can also make your relic weapon (The awesome bard harp/bow wep bard has in the artwork) if you level the right craft high enough though you can only get those after getting the mats and beating each primal in hard mode.
What else is there to do in FFXIV ARR?
basically what you listed, dungeons (Some can be pretty fun when you're just learning with a patient or new party.) mmm... Boss / Primal fights in normal and hard mode theres coil...... which is apparently a hardcore end-game dungeon with a bunch of floors (I've never been myself personally) Gathering, Crafting, Fishing... and apparently more is to come but thats all I know of for now... ^-^;;