My point about the key signatures is that you have to hear it more than anything because often the signature means nothing beyond which notes to play:
"one sharp in the key signature" for example can mean EIGHT different keys: G maj, a dor, b phry, c lyd, d mixo, e minor, or F# loc, or any synthetic key sig someone might have made up.
*Edit* You can always look at the first or last note of a piece, but does that say something? Check out the opening bar of Brahm's Rhopsody in g minor. It seems to start on the V in g minor, but the first full bar begins on an E-flat major chord, first inversion, or are those actually non-harmonic tones suspending the entire phrase to the first cadence in.... G Major?

Golly-gee. This piece happens to clearly end in g minor, though with a lot of emphasis on D major (if I recall), so we accept "g minor" in the title, although Brahms never said that--he just said "I have these here two Rahopsodies, published as Op.79."
"nothing in the key sig" can mean absolutely anything.
Also, Shubert's Impromptu D.899/4(?) "says" A-flat Major in the key sig, but it starts in a-flat minor, but it ends in A-flat major and that is how he wrote the key sig, so we accept "Impromptu in A-flat Major" as an accepted title.
Also, it doesn't have to be this complicated: Only very early music and modern music has stemmed away from two choices for each key signature, with few exceptions in between.
Regarding the counting of the time sig you presented, measures that can be counted in multiple ways utilize "compound time sig" because the counting in compounded, and Maryruth wrote out.
This confuses kids because it proves that 6/8 does NOT have to = 3/4 like they learned in math class! And it drives them NUTS! I had a kid tell me that he literally argued with a teacher and said that 6/8 = 2/4 because you count 6/8 in two, yet you reduce the fraction in math to 3/4.