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There are many types of euphorbia ranging from small succulents to large trees. They all have a different method of care. It would be hard to say without more information. If you could include some pictures, then that would help a bunch.

It is very likely that a cold snap will do this. They are only hardy down to 30 degrees, and below that will die off completely. Anything below 50, and the growth will slow down. Here is an article that will help: https://www.gardeningknowhow.com/ornamental/cacti-succulents/pencil-cactus/care-pencil-cactus.htm

Yes. This could, certainly, be mealy bugs... I would recommend watering with peroxide at half strength, and spraying with neem oil.

This article will give you more information on the mealy but, or scale bug: https://www.gardeningknowhow.com/plant-problems/pests/insects/control-plant-scale.htm

Growing zone for Gopher Spurge plants is zone 5 - 10. This however is a guideline and with the cold that you mention, that is not the normal for your area; your plants may not have survived.
If you see now new growth over the next few warming weeks, this is likely the case. Make sure you give the roots long enough before handing in the towel. It can take quite a while for roots to send up shoots.

This can depend on how patient you are!

Make sure you are adequately watering the new plantings.

Watering is key to success for your plants.

https://www.gardeningknowhow.com/ornamental/shrubs/plumbago/growing-plumbago-plants.htm
https://www.gardeningknowhow.com/ornamental/shrubs/acalypha/growing-acalypha-copper-plants.htm
https://www.gardeningknowhow.com/ornamental/flowers/spurflower/growing-spurflower-plants.htm
https://www.gardeningknowhow.com/ornamental/flowers/felicia-daisy/growing-felicia-daisy-plants.htm

Although not a true cactus, it is a very close relative of Euphorbia baioensis. These will root themselves back in no time. Just continue to care for it like you would a cactus, and it will recover and be just fine.

Have care when handling the sap. Most species are relatively harmless, only causing mild irritation to the skin. Some, however, are very toxic, causing burns and breathing issues from inhaling sap particulate. Just make sure that you wash your hands after touching this plant.