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This article will refresh you on the care.
https://www.gardeningknowhow.com/ornamental/trees/italian-cypress/growing-italian-cypress-trees.htm

These can be a little cold sensitive. Anything under 0F (-17C) will give them a burn as you describe.

Over fertilization can cause these symptoms, as well.

Unfortunately, so can overwatering!

There are multiple causes that could have caused this, and heat will not be one of them. These trees are tropical to temperate, and do best in warmer climates.

Regardless of temperature outside you will only want to water when the first 2 or 3 inches of soil is COMPLETELY dry. They will need regular irrigation, but if the soil is not allowed to dry out some, they will contract infections.

I would treat with a mixture of 1/3 peroxide and 2/3 water. Do this once per week for two weeks. This will treat anything growing in the soil that isn't supposed to be there.

Hold off on feeding for quite awhile. You will be able to get away with feeding a granular 1 or twice per year.

This article will give you more information: https://www.gardeningknowhow.com/ornamental/trees/italian-cypress/growing-italian-cypress-trees.htm

I can't speak to this brand of planter but a good rule of thumb is to use a container that is 2 inches larger in diameter than the nursery pot in which the tree grows. As the tree increases in size, repot it in a larger container until the tree reaches the size you desire.
Planting in the ground as long as you have the correctly planting location and soil will generally be better for the health of the tree vs a container.
https://www.gardeningknowhow.com/ornamental/trees/italian-cypress/growing-italian-cypress-trees.htm

If they are basically healthy and just need light trimming, then yes, they can tolerate a moderate height reduction and shaping which can be done aesthetically by a good tree service. But if you top or side prune/shape too heavily then the interior dry foliage will be exposed and the branches may not regrow and may die back more severely.

Browning can be related to water deficit, drought stress, but also Italian cypress are susceptible to pests and fungal disorders that can only be controlled with pesticides, or sometimes even that is not effective.

If the problem is browning from water deficit, then deep watering right away may save them. Installing irrigation and a compost mulch as you suggest, will benefit them and may stimulate new growth.

The problem with evergreens is that by the time damage can be seen in the foliage, it has been months maybe a year since the problem started. General information is that Russian cypress are known to be relatively free of common pests and diseases, although root rot is mentioned in some of the literature. They do like consistent moisture. Was there a dry spell last summer? What changes have taken place from the normal conditions?
You may consider having a local certified arborist inspect and provide diagnosis.

Was it droughty before the rain? Mites can cause problems such as this when summers dry. Here's more:
https://hgic.clemson.edu/factsheet/bald-cypress/

https://www.gardeningknowhow.com/plant-problems/pests/insects/spider-mite-tree-damage.htm