The headphone out is a TRS stereo but single 1/4 inch TRS plug, which carries left and right signal. This is not a problem, but the guitar amp might be.
1) NO MIXER NO ADAPTOR, Even with adaptor, it is just shorting it in the sleeve of the adaptor, Just short it at the insert, meaning, You just plug in a mono TS cable 1/4 and it will short the stereo and send it mono. I did it regularly when my wife was using a very lightweight 88 key Korg SP series, forget the exact model. Anyways, no 1/4 outs only 1/8 inch headphone out. We used a stereo 1/8 adaptor to 1/4 mono, which was basically doing the same thing shorting the stereo to mono. Yours is easier, just plug in TS mono cable and rock on. Some might suggest it could damage the HP circuitry on the keyboard, never had an issue and KORG was fully aware of this when they made the stereo HP out as the only output.
2) Here is the issue, putting that vey hot HP signal into your guitar amp will work, but the HP out is not an instrument level signal, as mentioned it has ability to drive relatively low impedance loads and some higher impedance HPs, but instruments are even higher impedance, hence the guitar amp was not made to handle this hot signal. Another however, it will work. I use a GK 2x10 Bass amp for my wife''s
NORD Stage 2 when she plays the role of bass with her left hand on keyboard with a Latin group. However, one more time, these are line level outputs, but again, lower impedance than pure instrument plug in. Sounds great and bass lines from keyboard with bass samples split with Fender Rhodes, but I can tell the amp circuit is not thrilled about it. Just seems a bit staticy, and unstable. However, last time, been 4 years of using it several times a week for the first few years and lately not so often. We do use it now as a Bass amp go figure, works fine, sounds great.
Good luck, no adaptor needed, even an adaptor as far as I know will not solve the fact that there is no output for the ground for the TRS on the mono end. Just shorting it is most common way.