It's not so unusual; I've seen amperage ratings in datasheets before.
Hm, I forget offhand which brands I've seen using them.
Another rough guess is this: find ESR at the required frequency. (If they don't have impedance and C(V) curves, keep shopping.) Assume power dissipation is half that of an equivalent sized resistor -- for 0805, that would be about 1/16W. The justification for this is, the ceramic material is lower thermal conductivity than the alumina substrate of the resistor, so the internal temperature rise will be proportionally higher. The temp rise also shouldn't be too high, as that affects capacitance, and probably has implications for mechanical stress (and eventual cracking). Put these two together, and that shall be your current limit: I = sqrt(P/ESR).
One or two amperes should be about right for 0805. With 9 phases, you'll have ballpark 1/9th ripple or 33A total, which should fit in the required area -- heh, but you may want to stack some things to get them in there
and the inductors
and the switches (2-sided load at least, but even multiple boards with board-to-board headers may be an option).
With the ESR, there is actual resistance and also dielectric loss. There can also be Eddy currents and the magnetic flield pushing the current to the outside - so the actal loss may be higher than the simple addition of the single components.
Yes, current capacity drops with rising frequency. This is a good reason to stick to smaller parts. I'm not sure where the cutoff is exactly -- you'd need a figure for the effective resistance of the capacitor (metallization and dielectric*) to calculate that. I wouldn't recommend anything bigger than 1210 anyway, for mechanical reasons if nothing else; whatever is probably fine at 1MHz.
*Yes, dielectric exhibits skin effect as well -- since it supports more propagation than loss, it's generally better to model it as, well, a lossy dielectric; but it works this way, too. The cool thing about physics is, it doesn't matter
how the loss arises, just that it's there -- anywhere there's an equivalent bulk resistance, you will find skin effect!
Analogously, this also matters in ferrite, but as the loss is modest, it takes quite a thickness to have much skin effect -- irrelevant in most electronics, but actually an important consideration in very large transformers, such as used in power transmission converters, induction heating, etc. Think 100s kW at 10s kHz, with cores the size of your leg.
(Which aren't standard parts, and good luck ordering them in such small quantities as these -- they're usually built from ferrite bricks glued together, which also conveniently provides air gap between blocks, permitting magnetic field to surround them, much as laminated iron serves for mains frequency transformers.)
Capacitor loss can be significant. I have seen water cooled mica caps in an induction heater. Not sure if really needed, as water cooling was there anyway, so it was low effort.
This one is easier to calculate -- simply take the total ESR and the total ripple currents (input and output), after all. The nice thing about bypass is, you can just use more capacitance, and the reactive power keeps going down. Reactive power being ripple voltage times ripple current. We can reduce voltage arbitrarily, and thus we can also afford to use quite lossy capacitors (type 2 dielectric). This is of great importance to resonant supplies (the active components need quite high Q), and other resonant applications of course!
Induction heaters usually have the double whammy of needing high power density as well as high reactive power, so it's almost impossible to get anything done without water cooling, at least above a few kW.
Tim