Yes, the Temple must be rebuilt before sacrifices can be offered up again.
No, we may not build the Beit Hamikdash. The Rabbis teach that the last and everlasting Temple will be the Third one and it is for that we pray and beseech God for every day, but it can only be rebuilt with the express command of God. The Holy Temple represented the closeness and the uniqueness of the relationship that exists between God and the Jewish People. That is why its destruction was such a calamity and that is why its rebuilding, may it happen speedily in our days, is so vital. With the advent of the Messianic era that ability to tap into the Divine relationship will be so smooth and the Third Temple will be a manifestation of that.