No problem for the HE400i. At 93dB/1mW it only needs roughly 512mW to hit 120dB peaks. It's 35ohms so no significant variation in power output vs the 32ohm rating. The only thing you have to worry about is whether the distortion and noise are very low, and I wouldn't suspect that amp to have a problem.
It could be enough for the LCD-X...if they were listing their specs correctly. Their homepage says 103dB/1mW, but all other sites list it as 103dB/1V, the latter being efficiency and not sensitivity. I think I remember elsewhere where the LCD-2 is listed to have 90dB/1mW, while Audeze's website lists 101dB/1mW but elsewhere is 101dB/1V. I forgot the equations but that's roughly around 93dB/1mW. Assuming that's accurate and power output doesn't drop below 32ohms (which sometimes happens, same way an OTL amp flips the power delivery profile but then trails off again above 300ohms), that's more than enough. However, if I overestimated the actual sensitivity figure and/or the output drops significantly below 512mW, then it's not enough for the ideal capability/potential to hit 120dB peaks. It will still get fairly loud most likely.
As for 600ohms headphones, you have two problems here. First, which 600ohm headphone? One may have 96one dB/1mW while another can have 99dB/1mW. That makes a lot of difference - one needs 256mW the other needs only 128mW. Second problem: you have no idea how much power it makes at 600ohms. Some older amps consistently halve the output when the impedance is doubled, but now you can have the power output not dropping that fast going from 32ohms to 300ohms. Then you have the Schiit Lyr and most AudioGD amps that will drastically drop from 32ohms to 300ohms, but drop the output at a lower rate from 300ohms to 600ohms. And yet, in some cases, despite not dropping that fast from 32ohms to 300ohms, it could drop at a much faster rate when you go above 300ohms. Assuming that at worst it might halve the output at 600ohms, then you only have 100mW. That will still get fairly loud on the 96dB/1mW headphone unless the gain is too low. Also, depending on its power delivery capabitlies, halving power isn't the only problem - it could also start piling distortion faster above 300ohms the closer you get to the max output at that impedance load.
You could try it and given it can work solely as a DAC, you can get the X7S later on to use with it. But given it's $365 you might as well just get something else - like the AudioGD NFB-11. It's got enough power for even the hungriest planars (which won't matter really as their sensitivity isn't as low as before), 300mW at 300ohms and 150mW at 600ohms; optional output for headphone, DAC direct fixed voltage, or preamp variable; and a ffront switch to select that output so when you integrate powered speakers (or a power amp with passive speakers) via variable output or an integrated amp for speakers or electrostat amp for electrostat headphones via fixed output, you don't need to pull whatever headphone is plugged into its front panel headphone output socket.
