The below answers are my opinions based on experience, and not taken from any official source. I strongly recommend asking your instructor for the authoritative source for the answers, which in many cases will be the POH.
max speed for moderate and severe turbulence? (flight manual gives me the recommended air penetration speeds but it doesnt say for severe or moderate turbulence) so not to sure..
Use the flight manual speed for rough air. If it's very bumpy (severe turbulence) use Va. Or even a bit less, to allow a safety factor. I personally would avoid known severe turbulence where possible, but if flown correctly there is no reason why the 172 shouldn't handle it. History has shown them to be rather strong.
what is the limiting factor for crosswind?
The ability of the pilot. Or were you after the value? 17kt, IIRC.
How did cessna arrive at their cross wind figure for the c172R?
Probably the same as any other light a/c manufacturer. Planted it uncorrected on the pavement, ascertained that the landing gear was more than capable of handling the side load, deducted a knot or two for the legal department, and wrote that value into the handbook.
factors that influence a particular figure to be nominated as Vne for an aircraft???
Includes the strength of the structure, flutter resistance of all the control surfaces, and the point at which redline RPM will occur in a fixed pitch a/c, with the throttle closed. Vne is the redline. As part of the certification, there is a safety margin in this, the actual speed the aircraft was flown to is, I believe, referred to as demonstrated dive speed (Vdf) which is about 15% higher than placarded Vne.
what is a trigger notam?
where would you find the table detailing when a SPECI should be issued?
Pass.
what is the max carb/heat drop in rpm? (havnt flown in a carb heated aircraft yet).
If the 172 is a carburetted a/c, this limitation will be in the POH. It's usually in the order of about a hundred revs. And a correction to the above poster: The heated air is less dense, not more, and it is the reduced density more than the incorrect mixture strength that is responsible for most of the rpm drop.
If it drops more than the permitted drop, it's possible that there is ice present. Leaving the carb heat on should very quickly restore the rpm to within limits. Or, the mixture is over-rich, or there is an obstruction somewhere.