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I think that your best bet would be to put it into a pot with damp potting soil. Keep the humidity around it high and keep the soil damp but not soaked. This may not work, but would be your best chance.

All of these are fairly low maintenance plants and enjoy spending summers outdoors in your area but must be overwintered indoors. Money tree does well in minimal sunlight and will not tolerate temperatures below 28 degrees F. Stromanthe needs to be placed in shade to filtered sun and cannot tolerate temps falling below 59 degrees F. Aaralia plants survive under low light as they long as they have moist soil. Indoors they do best in medium or full sun but should be planted in shady areas outside. Bring them back inside once temps drop beneath 60 degrees F. The ponytail palm, while easy to care for, generally needs bright light, but because it is such a forgiving plant, it will be ok if you give it bright light about half of the time. If you keep it in low light conditions half the year and provide bright light conditions the other half the year, it will be perfectly happy. This means that as long as you place it outdoors in the summer in a fairly bright area (no direct sunlight), it will tolerate any indoor light conditions you keep it in during the winter.

Yes, sounds like this would be a good time to repot. Here are some tips I hope you can use: https://www.gardeningknowhow.com/houseplants/ponytail-palm/ponytail-palm-care.htm

Sorry, you can't "shave" the sides. You need a pot large enough to accommodate the ponytail, so it will look attractive and balanced in its pot. The pot should be large enough to leave at least 2" of soil around the outside of the trunk. Also a shallower pot would look better, I think. You should be able to find another plant that would look handsome in that nice pot.

Ponytail palms are pretty tough, so it's not for sure that cutting the side off will kill it. It will definitely increase the chance for disease though. If cutting one small area will allow the plant to slip inside the pot rather than perch on the top, and you don't feel you can use a different pot, and you're prepared for the possibility of losing the plant, then go ahead and cut it. Let us know how its doing a year from now.

Since the ponytail palm is not a palm but is actually related to agaves, if you cut back that long trunk to whatever spot you want, there's a good chance it will start new growth, probably several heads, from near the cut end. You can also attempt to root the cut section. I would hold off on the cutting back until spring, when you can take the headless trunk outside (not to much sun if it's always been indoors) and water well, a little fertilizer, and the plant might well send out new leaves. It's not a sure thing, but a number of people have reported success doing this.