It can help if you write yourself out the list of sharps (F C G D A E B) and flats (B E A D G C F), then tick off the ones that are used in the chords for each one of your problems. Then go with the key that has that number of sharps or flats. A trick for the sharps: go up a half step after the last sharp on your scale, and that's your key. For the flats, go to the flat preceding it in the list, and that's your key (if the only flat is B, then your key is F). This works for the major keys only.
I'll do number 3 for you, then you can use the method to do the rest.
You have your list of sharps: F C G D A E B.
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GM is all white keys, so we can't check one off. (F C G D A E B)
AM has one sharp, C, so you can cross out that one. By default, as you can't skip ahead on the sharps list, cross off F as well. (G D A E B)
BM uses a D sharp and and F sharp (I told you F would be out:), so we can cross off D as well, and by default, G, since you can't skip ahead when you write them out on a staff. (A E B)
Em does not use a black key, so we can't cross one off.
GM - same as the first, so we can't cross anymore off.
C#m uses C# and G# so we're left with (A E B).
DM uses an F# which we've already crossed out.
That means that we use 4 sharps: F C G D - So it has to be the key of EM or the minor counterpart C#m....
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Which means that there might be a mistake in your question. If you have an Em chord, then you wouldn't (naturally) have a C#m chord. C#m uses a G#; so does EM. In a natural key, a note is static. It doesn't waver between being a sharp a flat or a natural unless the composer changes keys. There's a problem with the last one too. We've established that there is a D# in this key, but D major uses a D natural.
However, if you've given us your assignment exactly as it was given to you...then it's
EM with a minor tonic (i), or the first chord is minor and some kind of weird augmented seventh chord or it's
C#m with some kind of diminished third and a bunch more issues I can't even count.
If your teacher does not actually want you to map out the progression, but just to figure out the key based on the number of sharps or flats given in the other chords, then it's either EM or C#m.
If they want you to take the progression into the account - then this assignment makes no sense. You really should ask them - before I give myself an anneurism trying to make this make sense!!