Potatoes

No, leave them in the ground. Mine came back after the leaves were all burned by frost. They look great now.

Once the frost has got them they've had it I'm afraid, re seed within the next two weeks and fertiliser will be necessary as the first crop would have taken alot from the ground to grow that far.

It's best not to prune the potato plant in this case, because it needs its leaves to make food for the growing tubers. If some of the plant's leaves are pruned off, then the quality/size of the potatoes may suffer.

Once the stalks have died off.

Once your plants are 6 to 8 inches tall, hoe extra soil loosely around the base of the plant -- to within about 1 inch of the lower leaves from both sides of the row. Repeat in about 2 to 3 weeks. You "hill" the soil so that water doesn’t puddle around the seed. As the vines grow, mound up soil, covering the vine with just the tip remaining above the top of the mound. Do this until the mound is about 12 inches tall. Potato tubers develop above the planted seed piece, not below it. This is why tubers need to be covered with soil at all times.

Sad to say, maybe. Potatoes are defined as a cool-season vegetable. Your area gets so hot, it might be wise in future to avoid the peak heat with a variety that better suited to the climate; there are early, mid- and late-season varieties by planting in late summer or fall. Consult your local extension org or local nursery for specific advice. In the meantime, you could maybe try to keep them cool... If the mound is 3 feet high, it's also possible that you will get even a small harvest by/before June. We hope this helps.

Just go ahead and cut them up for planting. leave a couple of inches of the sprout sticking out of the soil. They will do fine. Protect them from frost with an old sheet if it gets cold.

How much rain? Potatoes like well-drained soil, ideally sandy. As long as it's not heavy and wet, you are fine. Excessive rain leads to tuber rot. Sandy soil dries out and warms up early in the spring, so that you can plant earlier.

Maturity time depends on the potato variety. Potato cultivars are grouped by maturity. Early maturing potatoes: 70-85 days after emergence. Medium maturing: 85-100 days after emergence. Late maturing: Over 100 days.

It's difficult for us to know if you will get potatoes, what with the conditions you describe (and don't include). It sounds like you have a microclimate environment—and that can be a good thing.
Just about any plant has a minimum soil temp tolerance. For potatoes, that's 40°F. A consistent 40°F. (Was that the soil temp when you planted??)
If additional cold conditions are expected, esp at night, you could cover the bucket. Use whatever is handy; the idea is to retain the heat that--presumably--the pot gathers in the course of the day.
Depending on how long/severe the cold is you might—emphasize might—be able to unearth the cuttings, bring them into the house in a portion of warm dirt and then replant later when temps are suitable.
Or you might have to start again. It's early in season; don't let that possibility deter you.
Good luck!

If the potatoes are deep enough in the soil and haven't started to show any growth above the soil they may be OK. If you are expecting more cold weather you can cover the basket with a blanket or a heavy sheet.

You may need to mound some more dirt around the potatoes as they grow taller. Put the stalks in between the rows to keep the weeds down.

If you planted seed potato pieces in a shallow trench, then the method is to plant with their eyes up (cut sides down). It will take about 2 to 4 weeks for stems to emerge with this method. You can always replant if it has only been 2 days. Just make a shallow trench (about 4 inches deep), plant, and recover with soil.

I start digging when the blossoms dry up, but you can carefully dig new potatos when the blossoms are full, but don't pull the plant up because there are lots of little taters still growing.