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A healthy hydrangea should be able to stand up to a rainstorm ok, which indicates to me that yours is "leggy" or has weak stems. Leggy plants normally happen when a plant is not getting enough light. While hydrangea like shade, they do need some shade and if they are in too deep of shade, they may develop weak stems. Another indicator that they are not getting enough light is if you get few or no blooms on the plant.

Another possibility is that the plant is not getting enough nutrients to grow properly and this is resulting in weak stems. If you feel this is the case, give the plant some balanced fertilizer.

The size of the pot depends on the size of the plant. You want it to be 2" larger than the rootball. Use potting soil, not topsoil, and there should be drainage holes at the bottom of the pot.

As long as you plant it in the shade, it should be fine.

Here is an article that you may find helpful: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/hydrangea-not-blooming.htm

It may not be the sun. They need a little sun, but not typically full sun. This article can give you some possibilities as to why yours is not blooming:
https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/hydrangea-not-blooming.htm

I'm afraid the best advice I can give is to wait. Since it is possible for these plants to come back from hard pruning (albeit delaying flowering), there is a chance that it may come back with some new growth. If not, you may need to get another and perhaps place some type of protective barrier around it to keep the dog away.