Well, they likely don't need one if their business is being done using their real name. so, Law Office of John Q. Public, probably can be done without a fictitious business name being filed, if the person running the business is John Q. Public. . Why do you ask?
I am sorry to hear all of this, and sympathize with the loss of a loved one.
Medical Malpractice cases are complicated, and even facts as detailed as these are not nearly enough from which to provide complete and detailed advice. Please do consult a medical malpractice attorney in your area who will be able to help you obtain medical records and get them evaluated to better understand whether malpractice has occurred, and what must be done next.
Not sure what you mean by "special appearances", since that term is usually used when contesting jurisdiction.
Perhaps some attorneys, but not many, will consent to only handling a given hearing, if the issues are isolated and straightforward.
Make sure to file a patent application first, THEN approach potential inventors.
Rarely do patent attorneys take equity interests in lieu of total compensation. Unfortunately, there are many, many inventions only a small portion of which turn into the next pet rock level of marketing sensation. However, I have heard of a friend taking a small interest as partial payment. This needs to be handled carefully.
An executor is a person who manages the estate and makes sure the beneficiaries get what they are supposed to get. You probably don't need to even mention a relationship to executors in the Will. Typically, family members might challenge who the beneficiaries are, and don't focus as much on who the executors are.
Make sure you use the proper wording and process for a self-proved will, if possible. Take a look here: