What to do with a stuck fermentation after 2 days with my first beer kit?

If after you check the gravity you truly are "stuck" the appropriate steps are to warm it up some and rouse the yeast.

To do this, I'd move the fermenter to a warmer spot for a day to ensure its a little warmer than where you were fermenting. If you can get it to 75F that would be fine. Then I would gently introduce a sanitized spoon or something deep into the fermentor to get to the cake. Gently stir it up some.

Another option is to never open it and gently swirl it around until you get the whole volume moving and the yeast cake starts to loosen up. I prefer that technique and have used it with good results. But if you are not in a clear carboy it can be hard to tell how well its going if you aren't used to it. Lastly, if you used a really flocculant English type yeast strain that cake can be pretty tight and difficult to rouse with swirling. Again the clear glass carboy is useful here.

You could try racking it to a new fermentor and suck up the yeast cake too, but that gets messy and increases the chances of oxidation and contamination.

Your first batch can be terrifying. You have nothing to compare to, and people are giving you conflicting advice. It's like driving at night with the lights off.... except...

This is beer. What you have there is probably beer, or well on its way, and not needing of your help. It's probably drinkable, or will be at some point. Here's some thoughts:

The OG reading you took seems low for the fermentables you put in (6 lbs LME, 1 lb DME). I put it through beercalculus.hopville.com because the kit doesn't give an expected gravity. I came up with 1.050. If you took your 1.031 OG reading while the wort was still hot, say 150F, this would correlate to a 60F reading of 1.050. If it was room temp, then that's puzzling.

Can you explain how you took the reading and how you adjusted it?

Most importantly: don't dump this batch. It's probably fine. I stressed out about my first batch, ended up drinking it early when I could've let it sit and mature. (It's 6 months old now and I'm left with one bottle, which is undoubtedly better than any of its predecessors.)

If you think it's underattenuated (taste for sweetness), add some dry champagne yeast and give it another week, then bottle and move on to batch #2.

To answer your second question, since you've had some fermentation, that means there's some alcohol in there, which means popping the top and checking the gravity is totally fine. Don't let your pet iguana crawl into it, and maybe don't be eating a messy sandwich over it while you're taking the reading, but your chances of infecting it by taking a reading after there's been significant fermentation are very, very low.

Before you do anythin take a hydrometer reading. The airlock is only an indicator and just because it is not bubbling it does not mean it is fermenting. ALWAYS TRUST THE HYDROMETER

I've had a batch of beer that fermented VERY quickly. As in finished fermenting within 48 hours of pitch.

1. Take a hydrometer reading. If it is near/at your final gravity, I'd say you are ready to bottle/keg. Once you are finished make sure the lid is secured properly and that the hole in the lid of the bucket is fully closed off. You may be fermenting and off-gassing, but due to a leak in the fermenter, not seeing any airlock activity.
2. If it's not finished, try aerating it. Give it a day.
3. If it's STILL not doing anything, pick up some yeast from the LHBS and pitch. Give it a couple of days.
4. If it's yet again STILL not doing anything, consider a sacrificing the batch to the beer gods and start over with a new batch.

You could try rousing the yeast and see if that gets it going. Grab your brewers paddle and give the wort a good mix up.