My greetings and welcome to Cyprus.
I was able to find this information which may be of some value to you:
Cyclist magazine did a very nice article on a recent trip to Cyprus. Would you have any way of finding a copy. This is the side of the guys who hosted them
I am sure they will be able to answer all your questions.
In general July and August are very warm (mid 30s to low 40s) so cycling from 11am to 4 pm is not advisable.
In terms of road cycling Troodos mountain is very good (nice scenery, smooth tarmac and roads not very busy during weekdays). The climb to Macheras mountain is also nice.
I am not into mountainbiking so I wouldn't be any help on that. The only thing I know is that there are numerous downhill and cross country paths and there are no restrictions on where you can go or what you can do. You can search for Cyprus downhill on you Tube and talk to the guys commenting under the videos.
Hope that helps