What company and/or government permits/licenses are required when selling an electrical device in the United States?
There are sort of two sides to this, a business side and a regulatory agency side. Business side, I don't know much about. I  would guess that if you need a business license, and don't have one, you won't get in that much trouble, and you can fix it by getting the license once you are discovered.

I worked for many product cycles at a well-known toy company. We had someone dedicated to regulatory issues, but I was a design engineer. Still, I am familiar with our practices. We generally did not UL test any of our products. The exception is the Lithium Ion or Lithium Polymer battery packs. I don't remember which UL standard they are tested to, but it was an important test because the turnaround time is very long and they need a lot of samples to perform the test. I would advise to avoid this by using off the shelf battery packs if possible, or use AA batteries.

For AC power adaptors we bought from vendors who had them tested to comply with a special toy standard for transformers. (They are not transformers... they are switchmode AC-DC converters, but "transformer" is the terminology used in the regulation). If you are planning to sell AC-DC converters, use off-the-shelf ones already designed and tested by people who specialize in that.

There are a lot of materials safety requirements on toys imposed by various agencies and also by the retail outlets. I would think that most of these would not apply if you are not in the toy market.

Then there is electromagnetic compatibility or compliance (EMC). FCC and ESD testing. Conducted emissions also if you plug into an AC power outlet. I am pretty sure you won't get in trouble for selling products that are not tested UNLESS they cause problems for other equipment. It is kind of irresponsible to sell something that is untested, but it is not clear to me that it is illegal (unless you falsely claim compliance).

I would say that most high speed digital circuit boards designed by amateurs and not tested will probably not pass EMC if they have headphone cables or USB cables or such. Boards that are physically small and have no cables often pass with no special effort. If you wire multiple boards together with ribbon cables, or add LCD screens or cameras to the board, then your chances of passing start to go down again. Wrapping cables around ferrite cores can help a lot. That is about all I can offer on this question, general as it is.

Oh, there is one other test we did, but it was an internal testing standard. We called it the "single short" test. Basically, you would short any two adjacent pins and make sure that there was no smoke and no fire and no external surfaces (such as the battery or battery contacts) got too hot. You sort of had to think through what would be likely to cause problems if shorted, and then come up with a testplan based on that. It was not a requirement that normal function resume when the short is removed. Many of the tests would result in board destruction. So this test consumed a lot of hardware.