Or is it characteristic of alkynes only ? also, can they (the alkenes) have double and triple bonds in the same molecule ?
A compound containing both a C-C double and triple bond is both an alkene and an alkyne. This is readily apparent in such a compound's systematic name, which will be something like alk-#-en-#-yne. Such a compound can be referred to as an enyne, or (equivalently) alkenyne, especially if the bonds are conjugated.
Alkenes can have triple bonds but, they should be between carbon and any other hetero elements and not between carbon and carbon. Otherwise, the compound ceases to be an alkene. Feel free to correct me if I am fallible.