Yes. It's a bit less common, possibly used more in the military and in technical discussions, but you do hear it. One note, though, an installation or facility doesn't have to be a large plant or power production site. A plant generally is where something is produced or made. This could be power or products of some sort: paper, food, cars, toys, etc.
Yes, but only as stand-alone words for variety's sake. In other words, in the US (where Cypherpunk and I are from), you would say or write "power plant," not usually "power facility" or "power installation." However, after having referred to a "power plant," you could then say or write "the facility" or "the installation" if you wanted to avoid repeating the word "plant." Or, you could say or write things like "This power plant is a small installation owned by . . ." The owner or operator of such a facility might refer to it geographically as "the North Anna facility" (a real example; there is a nuclear power plant on the North Anna River in Virginia) or "the North Anna installation" but in the United States the usual term for an unnamed industrial site where electricity is produced would be "power plant."