I'll take a shot at answering your questions the best I can...
Can I use an ale yeast or do I need to get a specific cider yeast?
Yes, you can use most any yeast, but you need to decide what you want out of your cider in order to decide what yeast to use. There's 3 things to consider when picking your yeast: Attenuation (how much sugar it will leave behind), Alcohol tolerance (how strong of a cider you can make), and yeast strain characteristics (what flavors/aromas the yeast will impart to your cider).
Step 1 Alcohol tolerance: decide how strong of cider you're making. Straight apple just will be around 5% abv, most any yeast (except bread yeast) will handle that.
Step 2 Attenuation: the lower the attenuation of the yeast, the more sugar will be ungerminated, hence the sweeter the cider; or the higher the attenuation, the dryer the cider.
Step 3 Yeast characteristics: look for a 'clean fermenting' yeast, something that won't leave a bread smell/taste, or possibly a yeast with more easterly (fruity) characteristics. Keep in mind that if the yeast is marketed for ale, the description of the characteristics for that yeast are referring to it's response to ale, not cider; a particular strain of yeast may impart certain flavors in ale, but not in cider.
My experience is that most people think they want a sweet cider (because they imagine alcoholic apple juice) and once they brew a batch or two, they realize they prefer a dry/bubbly cider (much more like champaign than apple juice). I would suggest using Red Star Pasture Champaign yeast.
What will happen if I do my primary fermentation in a Mr. Beer kit which has no airlock?
I'm not familiar with the MR. Beer kits. I can tell you that if it ferments beer, it can ferment cider. I'm assuming that by not having an air lock, you're not saying it's a sealed unit. It has to have some sort of vent for the CO2 or it would explode. The point of an air lock is to create a water/sanitizer/vodka barrier that keeps bacteria from entering the fermenter, but just having a small vent is enough.
I'll be using my only carboy for a batch of beer. Is there any way to "rig" some other container to go to secondary?
Any container that can be sanitized (preferably glass) and isn't oxygen permeable can be used. The trick is to find one that a airlock/stopper will fit into, or you can cover the top with a balloon, or even foil. The main idea is to vent the CO2 without letting in bacteria. The C02 escaping will prevent bacteria from entering the vessel while the primary fermentation is occurring, but if you don't use an airlock or balloon, you'll want to be careful to not leave it in that vessel for too long. Once the wort/cider stops producing CO2, there's nothing to stop the bacteria from entering.
I'm using a Mr. Beer concentrate. I'm planning on adding some juice. If I want to carbonate, at what stage to I add sugar?
Add 1 12oz. can of frozen apple juice concentrate just before bottling, and stir the crap out of it. Stir gently, so you don't aerate the cider. Or you could use 3/4 cup corn sugar, but the frozen apple juice concentrate will do just fine, and tends to add more apple flavor.
My last piece of advice is to read the following two links, in order:
1)
How to make Hard Cider
2)
How to Brew
EDIT: Sorry, I was thinking about 5 gal. batches. If you're doing a smaller batch (I think Mr. Beer dose 2 1/2 gal. batches) you'll want to scale down the priming/carbing sugar/juice concentrate. Otherwise you'll create bottle bombs.