Is it permissible to pray with more than one intention, such as the intention of fulfilling the obligatory prayer and of populating the mosques, and will one be rewarded for these intentions, and how should they be intended?

The "i'mar" (revitalization/inhabitation) of mosques is achieved by praying in them, even without a specific intention for that. If a worshipper intends the "i'mar" of the mosque along with the intention of prayer, this is not harmful, as it is a combination of two acts of worship, one of which occurs even without a specific intention. The manner of this is for the worshipper to bring to mind the purpose for which they are praying, such as intending the obligatory prayer and the greeting of the mosque ("tahiyyat al-masjid"). The correct view is that if one intends an act of worship and another act that is not worship, such as intending ablution for both purification and cooling oneself, this is not harmful, even if it diminishes the reward. The important thing is the validity of the intended "niyyah" (intention).