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With both carrots and onions, you can judge if they are ready for harvesting by digging out a little next to where the root is and see how big the root has gotten. If they are an acceptable size, then you can harvest.

With the onions though, once those leaves die (turn yellow) the bulbs will not get bigger, so you have to harvest at that time - regardless of size.

The following article should be of some help: https://www.gardeningknowhow.com/edible/vegetables/carrot/how-to-grow-carrots.htm

You can sprout carrots from their tips and even plant them; however, whether these will produce a 'bunch' of them, I do not know. I guess the only way to know for sure is to give it a try.

I would try watering both in the evening and in the morning.

You can also try putting a timer on your hose to water when you are not home. Hose timers are pretty cheap and can be bought at many hardware and big box stores.

You may also want to try using an opaque cloth draped over the area to keep the direct sun off the soil and seeds but still allow sunlight to get there.

You might find this article helpful:
https://www.gardeningknowhow.com/edible/vegetables/carrot/how-to-grow-carrots.htm

This article will help you with that:
https://www.gardeningknowhow.com/edible/vegetables/carrot/overwintering-carrots.htm

This article will help you: https://www.gardeningknowhow.com/edible/vegetables/carrot/overwintering-carrots.htm