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This happens when the soil is not loose enough. I would guess that your soil is either clay heavy or rocky, which makes it hard for the roots to grow downward well. If your soil is clay, try amending it. This article will help with that: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/how-to-improve-clay-soil.htm

If your soil is rocky, it would be a good idea to create a root vegetable bed, where the rocks have been sifted out or where the soil has been replaced with clean topsoil.

This article may also help: https://www.gardeningknowhow.com/edible/vegetables/carrot/fix-deformed-carrots.htm

No, you do not need to mound up the soil. This article will help with growing carrots: https://www.gardeningknowhow.com/edible/vegetables/carrot/how-to-grow-carrots.htm

You can try, it won't hurt to try. That being said, typically carrots do not respond well to transplanting.

If you are having issues with this pest, the following article should be of some help: https://www.gardeningknowhow.com/edible/vegetables/carrot/carrot-rust-fly-control-tips.htm

Carrots need loose, well-draining soil to grow well. The following article should be of some help to you: https://www.gardeningknowhow.com/edible/vegetables/carrot/how-to-grow-carrots.htm

Harvesting of the carrots can begin when they are finger size. However, you can allow them to stay in the soil until winter if you mulch the garden well.

To check the size of your carrots, gently remove some dirt from the top of the root and check the size of the root. To harvest, gently lift the carrot from the soil.