I want to learn this piece, but I have some questions.
1. Can it be played on a 61-key keyboard?
2. Is it suitable for my skill level? I've tried Nocturne Op9No2, so is it around that level of difficulty?

Thanks.

1) No, there's no serious classical music wich can be played on any keyboard. Get a clavinova or an accoustic piano if you want to play classical music
2) the 3rd movement is harder than op9/2. Btw, the first (famous) movement is technically easy, but musically quite hard.

How well do you want to play it?

Do you mean the first movement or all three?

I was hoping to play all three, but after checking all of them out on youtube (I had no idea that there were three movements. I've only heard the first one.) The third movement seems impossible, at least at my level, so i dunno...... Maybe just the first two?

Oh, and do any of you know any pieces that can be played with 61 keys?

I'm not sure, but I think Mozart "Rondo Alla Turca" can be played. 

OK, then.
Checklist:
1. Rondo Alla Turca
2. Piano Sonata No 19/20

Thanks.

I've always understood that the pianos Beethoven wrote his earliest works for had fewer than 88 keys. Was I wrong?

Beethoven had 2 and a half octaves above middle C first of all, then i think around 1803ish he got one with 3 octaves above middle C, which caused him to revise his 3rd piano concerto in C minor.

hey chong!
most of the repertory before the 19th century is suited to play with 61 keys. french baroque music, bach, haydn, mozart...the list is quite long. some early 20th century music is possible as well: bartok, stravinsky, turina.

of course, some subtleties are impossible and your technique (in a broad sense) will not develop properly. anyway, it's better than nothing, isn't it?

i would only say to get away from any piece that require something else, even the 62th key. furthermore, pieces that require precise and/or extensive use of the pedal you should avoid. then, beethoven's opus 27/2 is a very bad idea.

best!

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ps: sorry. about your actual question, i agree with people said: the first is way harder to render musically than the nocturne (and probably impossible on a keyboard - even a 88-key one), and the third is way harder to get under your fingers.

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pps: anyway...if you want just to have fun, and don't care about anything else, play whatever you like and be happy.