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Yes, remove the spent flowers down to the base of the plant.

Aloe Vera can actually get sunburned or sunscald when indoors all winter then being taken outside in to bright sunlight, it can be stressful and shocking for them to go from one extreme to another. Stick it in a spot with less direct sun and slowly increase the levels of sun it gets each day.

If the aloe pup (the second plant) had a complete root system on it, you could simply have put it in prepared dirt and let it root, given the proper care. Since it does not, I'm afraid the second plant will not survive no matter how hard you try.

This article will explain how to properly start a new plant from a mother plant: https://www.gardeningknowhow.com/houseplants/aloe-vera/aloe-plant-propagation.htm

you over saturated it, it is essentially a succulent. it needs almost no water. a good sprinkle of slow release granulated vegetable fertilizer every 6 mo. and a well drained pot with sandy soil if about all it needs. Treat it like an orchid, only water it when the bottom of the soil is dry then let it dry completely. If it does look dead though, just clean up the dead stuff and leave it be, it will come back. I have had two that have gotten frozen and died off above ground and then came back the next year.

You certainly can give it a try.

Use a small pot and repot with a cactus potting mix.

Keep the soil just moist; not dry or soggy.

Keep it in a bright location and hope for the best!

After the flower has faded you can remove from the plant.
https://www.gardeningknowhow.com/houseplants/aloe-vera/flowering-aloe-vera-plants.htm