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The ZZ plant will be healthier and happier growing in soil. While it can be grown in water, this is more of a novelty- it may be more susceptible to disease, and you will need to feed it with hydroponic plant food. In fact, they are happier with low amounts of water when grown in soil:
https://www.gardeningknowhow.com/houseplants/zz-plant/caring-for-zz-plant.htm

It appears you have a ZZ plant.
Your plant looks to be receiving too much sunlight.
Here is a link with care information.

https://www.gardeningknowhow.com/houseplants/zz-plant/caring-for-zz-plant.htm

Yes, you can move the plant up one pot size.
You can also prune back the plant and even propagate cuttings from the plant.
Here are some links with more information.

https://www.gardeningknowhow.com/houseplants/zz-plant/caring-for-zz-plant.htm
https://www.gardeningknowhow.com/houseplants/zz-plant/propagating-zz-plants.htm

Stop watering it frequently, ZZs need very little water. These articles should help
https://www.gardeningknowhow.com/houseplants/zz-plant/

The plant appears to be pot bound which doesn't account for the droopy stems, but it also appears that it may be overwatered as the stems have a grayish green colour at the bends. I would consider removing the damaged stems. Allowing the plant to dry out if soggy, and them transplant into a larger pot. Be sure to use a well draining potting soil at this plant doesn't like soggy feet and need little water. Also, is this plant near a air conditioning vent? That me make things worse.

This could be due to overwatering; the biggest issue with ZZ plants.
This article will refresh you on the care requirements.

https://www.gardeningknowhow.com/houseplants/zz-plant/caring-for-zz-plant.htm