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Yes, you can grow Rhododendrons in containers.

This article has more information.

https://www.gardeningknowhow.com/ornamental/shrubs/rhododendron/rhododendrons-in-containers.htm

If you have a non-native rhododendron, it has a shallow and wide-spreading root zone. The more of this you can dig up, the better. Have the new hole prepared before digging up the plant. Put the rhodie on a thick plastic sheet and drag it to the new location; this is easier on your back on leaves more soil attached to the roots. Make sure you plant it at the same depth or just a bit higher. Don't dig the hole deeper than necessary to prevent settling. Tamp the soil gently with your foot and water as you backfill dirt. Make sure to continue watering throughout the fall. More info here: https://www.rhododendron.org/transplant.htm

No, I would not advise this, as full sun will likely kill it. It is best planted in shade, so if you can find an area that receives, mostly, shade then that would be ideal. Here is an article that will help: https://www.gardeningknowhow.com/ornamental/shrubs/rhododendron/rhododendron-care.htm

The most common culprit in this problem is drought or inability of water to reach the roots. If the foliage has become dense, rain might not be getting through and the plant could benefit from being watered. Another possibility is insufficient light- if the amount of shade on the plant has increased since the last time it bloomed, this could be the explanation.

It will not hurt it at all! If it has been wet in your area, you may want to dab the cut area of the plant with peroxide to keep it from getting an infection. Otherwise it will not hurt them.

This article will give you more information on the plant: https://www.gardeningknowhow.com/ornamental/shrubs/rhododendron/rhododendron-care.htm