What key is this?

First of all, majorness and minorness (I'm not sure those are words) don't directly determine whether a chord fits into a key. They tell us which notes are in a chord, and the notes in a chord tell us which keys the chord will fit in. Both a C major chord and and a D minor chord will fit into the key of C major, because they both use only notes from the key of C. They will also fit into other keys, such as A minor. To determine which keys a C5 is in, just look at the notes in the chord (C and G, the 5 tells us that the chord contains just the root and fifth), and check to see if those notes exist in the key you're wondering about. Since *5 chords have only two notes, they fit into a lot of different keys, and I'm not going to try listing them. E flat major might be a good key for soloing, since those chords all fall within it if you look at them as flats rather than sharps.

It's called harmonising. If you look at the sequence of chords that harmonise with Cmajor (Cma, Dm, Em, Fma, Gma, Am, Bdim) and the notes contained in the chords, you'll see that they are all notes, contained in the Cmajor scale.
You can use the quality (major, minor, etc.) of the chord to help find the key a tune is in. There is a huge amount of music, which uses the I, IV, V chord progression, so that is a good place to start. In a major key, all 3 will be major chords (in C, they are C, F & G). In a minor key, they will all be minor - Aminor (the relative minor of Cmajor, with all the same notes as C maj) has the chords Am, Dm, Em.
The "5" chords are more difficult, because they are neither major nor minor - the 3rd degree (which is what determines major or minor) is missing. However, you can still start out by trying to see if the I, IV, V progression will fit.

Because power chords aren't really chords at all, but intervals, you can't analyze them as you do other changes. As Paul noted, they can be either major or minor - not because they have the qualities of either chord, but because they're missing the note that would determine the quality of the chord.

If you think of power chords as separate notes, you've got:

C5 = C, G
A#5 = A#, E# (or Bb, F)
G#5 = G#, D# (or Ab, Eb)

The enharmonic names that I've put in parenthesis are critical - because you simply don't see A# or G# chords very much. That's because they're in very awkward key signatures - the key of A# has 10 sharps (F, C, and G would be double sharps), and G# has 8 sharps (with G as a double sharp). Folks who notate chords this way don't understand theory, so they've just strung something together by ear.

Just because it's done by ear doesn't mean theory won't apply, and although most folks who come up with these A# chords think they're being 'creatively unhindered by theory' (or some similar garbage) doesn't mean the result is unique or novel... the opposite is usually true: by not knowing a lot of possibilities, the ones the ear leads you to are usually the simplest.

Back to figuring out the key:

The notes used in those chords, as sharps, are A#, C, D#, E#, G, G#. There's the first problem: two different G notes. Next problem: sharp keys have accidentals in a specific order: F, C, G, D, A, E, B. The last note in this series that you have sharped is E - but the C note isn't sharp. As a result, analyzing the series using sharps won't fly. Let's try flats:

The notes are: Ab, Bb, C, Eb, F, G. In flat keys, the flats appear in the exact opposite order as the sharp keys: B, E, A, D, G, C, F. So far so good - the three flats you have (B, E, A) fall in line, and there aren't any notes that would need to be flat because of these. Now we look for the first unflatted note in that series... you don't have a D note, but you do have a G, and it's not flatted. That means you can be in either the key of three flats (Eb) or the key of four flats (Ab).

Now let's see how the roots of the 'chords' line up in those keys:

Eb-F-G-Ab-Bb-C-D-Eb
Ab-Bb-C-Db-Eb-F-G-Ab

So.. in Eb, the progression would have the IV, V, and VI, and in Ab the progression would have the I, II, and III. Neither one is typical. Next you've got to use your ear to see how the melody falls against the chords - either C (if it's in Eb) or Bb and C (if it's in Ab) will have a more minor sound. Without more information, that's as far as I can go... but if it ends on C, I'm betting it's in C minor, using the i, VI, and VII chords.