May a commercial pilot transport an owner in the latter’s experimental aircraft for compensation? Say Olive Oyl owns an RV and wants to hire Popeye, a commercial pilot, to fly her to the beach in her aircraft, which has an experimental certificate. May Popeye legally accept compensation for this flight?
One might hastily conclude that the prohibition in FAR 91.319(a)(2) applies

§ 91.319 Aircraft having experimental certificates: Operating limitations.
(a) No person may operate an aircraft that has an experimental certificate —
     (1) For other than the purpose for which the certificate was issued; or
     (2) Carrying persons or property for compensation or hire.

but at least paragraphs (e), (f), (g), and (h) of the same section make exceptions that allow for-hire use. Well known for-hire uses in experimentals include ferry flights, instruction, test flights, aerial survey, and acceptance flights.
“Operate” and related terms have technical definitions in this context, provided in FAR 1.1.

Commercial operator means a person who, for compensation or hire, engages in the carriage by aircraft in air commerce of persons or property, other than as an air carrier or foreign air carrier or under the authority of Part 375 of this title. Where it is doubtful that an operation is for “compensation or hire,” the test applied is whether the carriage by air is merely incidental to the person's other business or is, in itself, a major enterprise for profit.
Operate, with respect to aircraft, means use, cause to use or
authorize to use aircraft, for the purpose (except as provided in
§ 91.13 of this chapter) of air navigation including the piloting of aircraft, with or without the right of legal control (as owner,
lessee, or otherwise).
Operational control, with respect to a flight, means the exercise of
authority over initiating, conducting or terminating a flight.

In the scenario above, are both Olive Oyl and Popeye operating the aircraft simultaneously? Who has operational control?
Interpretation of FAA regulations seems heavily weighted in favor of protecting the ignorant public from the general risks of flying and especially from unseemly outfits who may find irresistible the temptation to cut corners. Owners and those who otherwise provide the aircraft that will carry them or their property have different incentives and receive different treatment, cf. FAR 91.319(e)(2) and FAR 91.319(g).
If Olive Oyl asserts that she is not running a commercial operation and is paying Popeye for his services, do they have a legal leg to stand on?
This one seems pretty straightforward to me. 91.319(a)(2) applies (because in the scenario you have described the pilot would be carrying persons - Olive - for compensation or hire). 

Popeye could accept compensation to reposition the aircraft (with no passengers or cargo onboard) as a ferry pilot, but not to operate the aircraft for the purposes of transporting persons or property (besides the aircraft itself and the incidental things it might carry like extra oil) - even if the person happens to be the owner of the aircraft.