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Yes, you can prune out the affected portions of the tree. Also, I would treat it with neem oil. It is effective for ridding the plant of pests, but is still safe for people, animals, and even beneficial insects, like pollinators. This article will give you more information:
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

They should start out green. It sounds like the soil is lacking in nitrogen. Have your soil tested and then add some nitrogen rich fertilizer.

You can prune to make it bushier. While not essential, thinning branches allows the tree to have stronger branches and better support for fruit, as well as better airflow and sunlight penetration, which results in bigger fruit and a healthier tree.

They can survive outdoors in areas that do not get colder that -7C. If you live in an area that gets colder than this, you will need to grow it in a pot indoors or in a greenhouse.

Unfortunately, we cannot recommend anywhere to buy one.

It can recover from this, and as long as the stems are still alive (which should be the case unless it was a prolonged freeze), the leaves will regrow. This article should help with the revcovery process: https://www.gardeningknowhow.com/plant-problems/environmental/tips-for-saving-cold-damaged-plants.htm

This may be a sign of a pest infestation. Typically citrus leaf curl is caused by aphids but could also result from scale. The sappy substance is honeydew. Neem oil should help with the problem. Here is an article that will help as well: https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm