Does Egg Laying Cause Molting in Finches?

Removing the egg will probably result in more eggs being laid. Her chest is bare because she "feathered her nest". They should grow back. If you get her a mate, they will raise babies. There is nothing more cheery than a flock of finches.
Make sure she has a calcium source. Cuttle bones are great.
She will enjoy sprays of millet. You can grow your own bird food by planting some of your bird food. A leaky aquarium makes a great green house. Cover it with a piece of glass or a sheet of Saran wrap or other plastic to keep the moisture in and put it in a place that gets sun.

Transplant the plants outside when weather permits. Check with your local County Agricultural Agent or your state Department of Agriculture for more help.
Find a good finch husbandry website. Check out the library. Check out local bird clubs and breeders. There is a lot of good information out there and a lot of misinformation. Look up the AAV - Association of Avian Veterinarians. AAV vets are usually pricey, but they know a lot more about birds than the average bear.