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These articles should help: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/prune-hydrangea-bushes-hydrangea-pruning-instructions.htm, https://www.gardeningknowhow.com/ornamental/flowers/roses/pruning-roses.htm

Typically, in hydrangea, this is caused by a watering issue, though it could be a number of other things--over fertilizing, leaf scorch (if planted in too much sun), etc. Check the soil and make sure that it is damp, but not soaked. It should have drainage. Has it recently been planted? If newly planted, there may be gaps in the soil around the root ball that is drying out the roots. I would tamp down the soil around the plant. I would also recommend upping the watering. In addition, it could be suffering from shock. This article will help with that: https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

You can trim off the brown parts but it may be best to wait a little while to make sure there are no more cold snaps. This article may also help: https://www.gardeningknowhow.com/plant-problems/environmental/tips-for-saving-cold-damaged-plants.htm

I would wait until all threat of frost has passed first and then trim off only what is dead. They will regrow their leaves as long as the stems are alive. Most hydrangea bloom on old growth, so if you prune too much, you will lose the blooms next year. This article will help with pruning: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/prune-hydrangea-bushes-hydrangea-pruning-instructions.htm

It could be a few things. One is mites. They are too small to see, but do cause damage to hydrangeas. Another is a fungus. Again, there are several kinds, blight fungus being the most probable cause. Fortunately, there is an organic solution that treats both mites and fungus (as well as many other pests that could be affecting the shrub). Treat the affected areas with neem oil once a week for a month. Here is more information on neem oil: https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm