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Garlic water may help to repel them. Is the whole green top gone, or are you finding them laying near the plant. If the whole top is gone, it may be a rodent causing the damage. If you are finding the sheered off top, it might be cutworms. These articles will help with either issue: https://www.gardeningknowhow.com/plant-problems/pests/animals/garden-mouse-control.htm, https://www.gardeningknowhow.com/plant-problems/pests/insects/get-rid-cutworms.htm

Watering questions can never be answered by a "how often." There are many variables that affect water, such as type of soil and weather. Some digging beside where your beets are growing will let you know if the soil is damp all the way down to below the roots. Deep and thorough watering is the way to go; generally an inch per week will be sufficient, unless you have sandy soil, which may need watering twice a week or more. And obviously if there's rain you won't need water. This article has more beet growing information. https://www.gardeningknowhow.com/edible/vegetables/beets/care-of-potted-beets.htm

Usually, if you are getting all kinds of top growth and not much root growth, the soil or fertilizer is way high in its nitrogen content. Not enough food is available for the root system development. If you are using a high nitrogen fertilizer, I would cut back on its use and find a fertilizer with a more balanced N-P-K formulation, or one higher in phosphorus (bone meal is good).

If the leaves are "lace-like" that suggest something that is eating the leaves, like beet armyworms. https://www.gardeningknowhow.com/edible/vegetables/beets/beet-armyworm-control.htm
Some causes of fading and drying leaves could be water stress (too much or too little,) temperatures too warm or too much sun,and improper fertilizing. Here's some info on growing beets: https://www.gardeningknowhow.com/edible/vegetables/beets/growing-beets.htm

Yes, they are okay to eat. A very large beet will be more dense and somewhat less tender and sweet than a small one of the same variety. However, you can maximize their flavor and sweetness by roasting them.

Slugs can level tiny tender beet shoots completely. I would recommend looking for them, with a flashlight in hand, at night.Here is an article that will tell you how to deal with garden slugs:

https://www.gardeningknowhow.com/plant-problems/pests/insects/kill-garden-slugs.htm

As for rabbits - how high up is your raised bed? It is recommended to build a bed that is at least 18 inches high to discourage rabbits. Here is an article outlining some strategies to keep rabbits away from your garden:

https://www.gardeningknowhow.com/plant-problems/pests/animals/keep-rabbits-out-of-gardens.htm