I'll describe what I've seen at (traditional) Catholic masses. At some masses, the priest will consecrate a large number of hosts, more than would be needed at that mass. These are in a ciborium on the altar (See: Wikipedia for pictures of some ciboria), which is uncovered at the time of the consecration. They are consecrated along with the host that the priest will consume at that mass. Those that are left after communion are kept in the ciborium (covered) in the tabernacle on the altar, and they are available for distribution at later masses. I'm not aware of any particular rules about when additional hosts can be consecrated, and I would guess it's just whenever the priest notices that the supply in the tabernacle is getting low.
The preceding paragraph describes what I've seen in typical churches. In places where there is no church and the mass is celebrated in rented halls, there is no tabernacle in which hosts can be stored for the weeks (or more) that can elapse between masses. (Actually, hosts should not be stored for that long anyway, because they shouldn't be allowed to get moldy.) In such a situation, I've seen them count (before mass) the number of people planning to receive communion, so that the priest can consecrate the right number of hosts (maybe a few more than the count, in case of latecomers). If any hosts remained after mass, the priest would consume them.
If, for some reason, there are not enough hosts, the priest can break them and give each communicant half (or a smaller fraction) of a host. Any small part of a consecrated host is still Christ's whole body, blood, soul, and divinity (as long as the part still has the accidents of bread --- not just a molecule or two).
To answer your other questions: At the elevation, the priest elevates only the host that he will consume. The rest remain in the ciborium. The ciborium is covered except during the consecration and the distribution of communion.
Consecrated hosts are kept in the tabernacle, except during a special ceremony from Holy Thursday to Good Friday. Unconsecrated hosts are, I believe, stored in the sacristy, the room where the priest puts on his vestments and generally prepares for mass.