Where are you stuck?
Note that the first question is false as stated: Consider as a metric subspace of and let be a rotation about the origin of then has no fixed points on but is the identity.
Question one, as I showed you is false as it was stated, you need to ask the fixed point be unique.
For question two, notice that a strictly increasing function always has a left and right limit (even if they do not coincide) so the only discontinuities it can have are jumps, but the image is an interval.
For five, you need that composition of unif. cont. functions is again unif. cont. and the following fact: If is continous and bijective and is compact then is continous (to prove this use sequential compactness of ) so and you're done.
While you are still online, could I get your input on the following question, out of interest?
Let f be strictly increasing on a subset S of R. Assume that the image f(S) has one of the following properties:
(a) f(S) is open
(b) f(S) is connected
(c) f(S) is closed
Prove that f must be continuous on S.