The easy question first: receiving any money for doing so is illegal compensation. Sharing expenses doesn't apply because you don't have a common purpose with the patient.
It is likely that such an operation would require a part 135 certificate, since the business you are describing could be considered to be holding out a willingness to transport people for compensation. But even if you have a commercial certificate and think that it should be legal under part 91, 14 CFR § 91.319(a)(2) forbids it in an experimental aircraft:
No person may operate an aircraft that has an experimental certificate [...] Carrying persons or property for compensation or hire.
The FAA interprets "compensation" very broadly. If you receive anything of value, even if it's abstract or less than your expenses, you must fall into a specific exception that allows it or you are receiving illegal compensation. There are many exceptions for certificated aircraft, but for experimental aircraft there is only glider towing and flight training.
Otherwise, I think it should be legal as long as you follow all the other restrictions of 14 CFR § 91.319 and of your airworthiness certificate. Some of these might be a problem:
Unless otherwise authorized by the Administrator... no person may operate an aircraft that has an experimental certificate over a densely populated area or in a congested airway
Each person operating an aircraft that has an experimental certificate shall [...] Advise each person carried of the experimental nature of the aircraft [and] Operate under VFR, day only, unless otherwise specifically authorized by the Administrator
The day VFR and densely populated area restrictions are theoretically problematic but can be waived on your operating limitations. So in practice this doesn't really matter.
Needing to inform the patients that you are flying an experimental aircraft is probably more problematic. I expect the reaction of most people to being told that they are being flown in an "experimental airplane" is "hell no I'm not."