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If we could irradiate a metal or a plasma with a coherent beam of hard X-rays, would the main effect be an ordered oscillation of the electrons like with a classical EM wave, or "compton-style" collisions?

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You may find this (paywalled, unfortunately) reference useful: X-ray quantum optics, Adams et al., J. Mod. Opt. 60 p. 2 – Emilio Pisanty Sep 20 '13 at 16:33
Note that coherent x-ray sources are few and far between. The only way I know about to create them requires a nuclear explosion, but that may just be a statement of my ignorance. Are there less destructive terrestrial sources? – dmckee Dec 19 '13 at 22:48
Also note that X-ray lasers were the centerpiece of the crazier part of Ronald Reagan's "Star Wars" program, where, theoretically, space-mounted X-Ray lasers were going to locate missiles at the point of launch and destroy them before they had a chance to leave Soviet airspace. I'm unaware that there has been any further research into them since Star Wars was scaled back to the SDI system that is currently deployed. – Jerry Schirmer Dec 19 '13 at 23:56

Phase-contrast imaging with x rays has been achieved at the LCLS. This is used to probe extreme matter states, such as the ones created by a shock launched by high-intensity laser-pulses incident on metal slabs.

See this or this for instance.

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Ah! I had heard people were working on x-ray band FELs, but hadn't heard they'd been successful. Very cool. – dmckee Dec 20 '13 at 4:33

For example, When copper metal is irradiated with X-rays, some X-rays penetrate into metal and get absorbed. If the metal is thin, some X-rays come out from the other side with low intensity.

If the incident X-ray beam is at an angle to copper block, scattered X-rays can be detected. Depending upon angle of incidence, some back scattered X-rays are also expected.

Interestingly, if the incident X-ray energy is of higher energy than K X-ray energy of copper, characteristic Cu X-rays are emitted from copper block along with back scattered X-rays.

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This answer fails to address the "coherent" part of the question in any way. – dmckee Dec 19 '13 at 22:46

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