It might be a small spot of rot, but I've noticed that plant-killing rot is usually invisible, starting on the interior. Yours looks like it's isolated and the plant will probably seal itself. If it has roots and leaves, I'd give it a little water.

If it's soft it probably rot. However there's no need to be too concerned. Just may sure it doesn't spread by only water the soil. No misting etc.

I think the roots look good from the pictures and I would not get too wrapped up on this one spot at this point. IMO J and Andrew offer good suggestions.

My plumeria suffered a cold night while I was out of town immediately brought it and it's sister plant so thankfully it was only one night but this one is struggling to bounce back cut the dead parts off to where I thought looked healthier I was scared to cut any more

It had such beautiful pink flowers last summer I don't know what else to do for it

Generally for cold related plant tissue damage. It's recommended you don't cut anything off until it dies back by itself. Sometimes you risk cutting off parts that look damaged but are still viable. As for what you can do, give it plenty of light, warmth, and humidity. Plumeria thrive in these conditions. It may take.a while for it to recover from the damage even under these conditions. Best of luck!

Based on the picture It looks like the one on the left is in dire straits. Generally I agree with Andrew about letting a plant harden off the area when dealing with frost on a mature plant. In this case since this plant is quite young I would remove all the black areas you can. Just maybe the roots and enough of the trunk survived that you may get some regrowth.

the larger one looks ok to me from the picture provided but a more detailed exam might be warranted.