Are all or most of the hot Jupiters (exoplanets) found orbiting close to their parent stars assumed to be gas giants? Is there anything else they could be, such as terrestrial planets?

The default expectation is usually that they are Jupiter-style words (whether or not "gas giants" is the best name, given the amount of liquid and molecular material believed to make up the interiors). The only current way to test this notion is in the case of transiting planets, were we have a pretty good idea of both mass and size. For a given composition and temperature, we expect a certain diameter-mass relation. There is interesting scatter about this for some "hot Jupiters", although not enough to allow fully terrestrial-world composition. The high equilibrium temperatures certainly complicate this, although existing calculations take this into account as best we can at this point.

Any planet comparable to Jupiter in mass will be able to grab and hold onto
lots of gas. Although most of that gas will liquify under the pressure of the
gas above it.

-- Jeff, in Minneapolis

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"The other planets?
Well, they just happen to be there, but the point of rockets is to explore them!"
-- Kai Yeves