Ditto.

Covering them up with grass clippings sounds potentially very bad.

The soil may need help as said.

Can you tell if some animal has nibbled the tops?

Needs good sun to grow.

I did not realize potting soil was sterile, in fact I thought I had nutients mixed in it already. I heard not to put manure on carrots as this will cause forking. That's what I use for my other veggies, aged horse manure. What should I use for the carrots that's not chemical fertilizer?

Excess nitrogen causes forking.

Add just a little horse manure, water it in, and see how it goes.

Being sterile means it has nothing living in it, like bacteria. It does not mean it has no nutrients, since nitrogen, potassium and phosphorus are not alive. Potting soil often does have fertilizer mixed in. Carrots are notoriously slow, at least where I live. How fast they grow depends on light and temperature also.

I googled "living soil' and found this:

Millions of small organisms live in healthy soil which is rich in organic matter. Some of them are big enough to see, such as earthworms and small insects. But, most of them are so small that you can only see them if you use a microscope. These microscopic organisms include nematodes (tiny worms), bacteria, fungi and even some insects which are invisible to the naked eye. These living organisms make soil alive and give it a good structure and texture. A living soil ecosystem nurtures and nourishes plants by providing a healthy medium to take roots and through a steady supply of nutrients.

Q. how does one add-in all this to make a thriving pot or raised bed garden?

Matter of debate on how to organically help new or container gardens, and obviously people will try to sell you stuff. You could try worm castings or "compost tea". Many organic fertilizers claim to have additives to help, like bagged Espoma and Dr. Earth, both which now claim to be "organic" and add living microlife.