I was about to compose my third movement of my first sonata, but the other movements are notated Andante and Adagio. I would want my final movement to be notated Allegro or something even faster than Allegro, but can't seem to compose fast music. Is there anyway I could compose faster music without being a virtuoso pianist? Or I have to become one to compose this kind of music?
The simple answer is no. Think of it this way: Does a composer write pieces only for instruments he can play? No, he does not. He might compose pieces for instruments that he has never touched in his life.
You don't have to be a virtuoso pianist to compose fast music. You have to learn how to imagine what you want to compose. If you can imagine your piece at a fast tempo, you can write it down. You should be able to hear it in your head, but if you cannot, nowadays you can easily download a DAW that can play the piece for you and see how it sounds (if this is your problem). You can write the notes and the DAW will play it back for you (The sound outcome won't be like a real pianist playing the song, but it'll help you understand some parts if you cannot figure out how they sound exactly).
I think it is a common misconception that playing fast makes you great at your instrument. Playing slow music correctly and in time, with good interpretation is to me much more of a skill tester than playing fast.
Many times when there is music played a high tempo the speed takes center stage and a lot of the principles of good melody takes a back burner, but because it is played fast the somewhat boring melody gets masked by a million notes being played in a minute.
To the OP though. I don't think you need a certain skill set in a instrument an instrument to compose for it. You do need a good understanding of it. This understanding can come from playing the instrument well, but not always.
I think it fair to say many of our great composers over the ages did not play every instrument in the orchestra, but they still understood how they worked. They definitely used this knowledge in there compositions.
So no, don't get too hung up on how fast you can play the piano. Compose good music and let the quality of the music speak for itself.
You should be able to hear it in your head
That, IMO, pretty much wraps up what composing is all about: imagining how something would sound. Of course, one way to get this imagination is to actually listen to different possible options, by trying stuff on an instrument or a MIDI editor. But that's totally not necessary! In fact I'd go as far as saying it can be counterproductive to immediately play/listen to every detail as you write it: this often distracts from the grand concept.
So, if you have some melodic ideas and think they would sound good in a fast tempo – just write them down as such! It doesn't matter if you can play any of it in the proper tempo.
OTOH, you shouldn't just write something unplayably fast either just for the sake of having a fast movement. If the way it feels natural happens to be in a tempo you can also play yourself, then all the better.
Anyway, especially for a solo piece the choice of tempo is largely up to to the performer. You should certainly have a rough idea of how fast the piece should be, but a good composition will usually sound good in quite a range of different tempi. If you can yourself play it in a slow tempo and it sounds good, then chances are a virtuoso pianist will be able to play it faster and it'll sound even better – unless a fast tempo specifically contradicts the mood of the piece.
A bit of an edge case is if you can play everything slow and each individual movement sounds good this way, but the whole piece gets boring like that. In this case it's possible that speeding up one movement will to the trick to make the whole thing a round entity, but I wouldn't rely on it: boring and fast can also be just boring and annoying. Hence again the imagination bit: if you can imagine it in fast tempo and it feels exciting, then it probably is good. If you can't imagine it, then it probabably needs some more thorough thinking.
No, you don't have to even play an instrument to write for it. But you need to know how to write for that instrument, otherwise you can write things that are not playable, or that don't work well on the instrument.
Since you are asking the question, you don't know well how to write for piano, therefore I suggest you get advice from a good pianist who can play and try your ideas on the piano. That should help you.
I would say no you may not be. However you should have a very clear idea of what the music sounds like in your head. Composers who composed symphonies and so on were not able to make play or replay a whole orchestra just to arrange their music. Their minds were playing it.
No; you can compose music for an instrument or for vocals without you even knowing how to play that instrument or sing it for yourself. However, you should have a feel of the piece before you finalize, so that it could perfectly fit into the whole composition.
To accomplish this without hiring a virtuoso pianist, try free phone apps and place the notes/pitch in the track staff lines of the apps and change the instrument as you wish and hear the real music on the phone app. (Most MIDI with music notation apps support this feature on phone and tablets, for example Music Tool Kit from Google Play for Android phones.)