Often when the same sound is repeated in japanese, the second syllable is softened. I believe this is because it is easier to say, as in it rolls off the tongue easier. Obviously this doesn't apply when the first sound is already "soft" e.g. iroirona 色々な
There is no singular or plural in japanese. But "tokidoki" means that watching a movie happens more than once. But then again I have translated it as "I watch a movie once in a while" and I think that would also be a correct translation.
It is both present and future. Depends on the context of the sentence. If say あした or some other time word that indicated the action was set to happen in the future was used in a sentence then it would be clear that the verb would be translated as the future tense.