There are two separate terms:

A combat plane is a plane like the Lanier Ripper that is used for arial combat with other combat planes. There are streamers attached to the tails of both planes and the object is to cut the streamer off of the others tail.

A warbird, is a replica of a military airplane (like a P-51D Mustang). From just about any era.

No planes shoot anything out. Some drop bombs, but they don't shoot.