Is completing the construction of an existing mosque rewarded the same as building a complete mosque, or is it better to build a new mosque? And is the one who contributes to completing the construction of a mosque rewarded for building it?

It is better to build an independent mosque in a place where people need it, so that one may receive the full reward. If one is unable to do so, then they may participate with others, and their reward will be commensurate with their contribution. The need of the area for the mosque and the greater benefit of its existence must be considered, for the more beneficial the mosque, the greater the reward for its builder.