1. I have almost no idea... I'd assume that you do... can you call the embassy...?
2. Which Airport? If it's the San Jose one, staying near is pretty easy. San Jose itself is pretty small. Stay in a place where someone might be offering a room. Tico's (Costa Ricans) are really nice, and will almost always offer you food, hospitality, or any other form of kindness. I don't really know the names of the districts or anything, there's a market called HiperMas that has everything you'd ever need...
3.Arenal is one. Arenal used to be a mountain, so there was a town built up next to it, and when my abuela was a little girl, she was taught that it was a mountain. Until it erupted. The whole town was covered in an ashy substance, causing them all to go to sleep, then the lava came and ... you know... lava...
To this day, if your on the boat where the lake is, ask your guide to maybe point this out for you, you can still see the tops of buildings.
Oh, and there is an urban legend that there was a boy found wandering the rubble days afterward. I don't know much about that...
The second place is one of which I think you should go to is one that I forget the name of. It's a little out of the way of San Jose, it's beautiful, past a bunch of mountains and greenery. There are small little villages hidden in between them that are practically impossible to get to, a delicious restaurant or four, and a million-dollar view that you can't miss. Keywords: Dam, sky meets horizon, waterfall, lady who sells corn, small offering spot (deceased person...)
And the third place is the cloud forest. The cloud forest is so freaking humid and dense, but once you get out and start walking... look around... it's beautiful. Tree upon tree covers one another, everywhere you look there's an exotic insect, a weird-looking frog, maybe even a sloth if you're lucky. There's a spot inside of it called the hummingbird garden. The only man-made things there are the wooden benches and the hummingbird feeders, which are actually kind of unnecessary if you ask me. There are also a ton of butterfly's in the hummingbird garden, colors that you find only on parakeets... I really don't know whether it's called the hummingbird or butterfly garden, but they're both there in the same spot, anyway...
Oh, and a tip... Try not to ask Tico's for directions. You'll end up getting this: "Turn left on the place where the old church used to be, then go up the road where the steps to nowhere face left" Really. Get an extensive map.
^_^ I hope this helped!!