While Utgarde Pinnacle is very close it's unlikely to be opened as only 2 of the 4 bosses are working well.
Ahn'kahet isn't properly enabled for heroics yet so it won't be available (you'll get a message that you need to be level 81 to enter).

That leaves the following 5 man heroics:

• Utgarde Keep
• Azjol Nerub
• The Nexus
• Drak'tharon Keep
• Halls of Stone
• Halls of Lightning

Naxxramas is close enough that I'm going to open it, the only real way to figure out what needs fixing is to go in there as a raid.

The Tabard championing system isn't implemented yet so unfortunately you will need to keep doing dailies to get faction rep.

This update includes a large number of core code changes, supposedly fixes, but experience has shown us there will be a large number of regressions as well. So now you're warned.

Similarly there are a number of database changes, including damages. So don't be surprised if some mobs start hitting too hard or soft.

Finally on a more philosophical note: a lot of things don't work right here, bugs are fixed as time permits, tickets are handled when possible. If you play here you should accept these problems, however if you find this unacceptable I suggest you will have more fun by paying your subscription to the real thing.