I'm still wondering about a roof leak, especially with the flies. In a cabin with a metal roof, we had a problem with flies getting inside. In this case, the roof had not been properly installed. After having the roof re-installed, the fly problem was back to normal (opening a door lets flies in). Poorly sealed windows or other cracks can also let flies in, in which case the two issues of odor and flies may not be related.
It was wise to try a methane detector, but unless it is a good quality product, it may not be reliable. Either your building maintenance manager or the supplier of your natural gas (city utility company?) will have a qualified, reliable device. It would be good to have your apartment checked for a methane leak at a time when the odor is strong. Pungent sulfur odorants are added to natural gas (and propane) for safety to reveal leaks. Those odorants are a little different from the hydrogen sulfide odor, but if you aren't familiar with the different odors, it may be hard to recognize and describe. Methane has no odor and could come from either a sewer gas leak or a natural gas leak.
Because you are in an apartment, I wonder whether the peppermint oil test may not have worked as expected. Perhaps one of your neighbors enjoyed a nice unexpected mint aroma on that day. Because this aroma is strong, pervasive and lingering, it would be difficult if not impossible to come up with a way to insert oil on the back side of a trap without introducing it to the inside of your apartment.
The city water & wastewater department may be able to measure if hydrogen sulfide is present. That may require a request from the public health department, and the person sent to measure should be able to come out at a flexible time when the odor is strong. If there is no natural gas leak as determined by the gas/utility company device, then the source of the odor may be sewer gas.
By this point, I assume you have eliminated all common issues, such as the odor inside a front-loading washer when the drain port isn't regularly cleaned, or a garbage disposal, or a nearby livestock feeding operation. At a laundromat in our area, there is a smell like this coming from two sources: a very large washer that isn't used often and the restrooom trash can. (We drive a long way to wash comforters.) If the apartment below you or beside you doesn't have the same problem, and you can rule out a natural gas leak, a sewer gas leak seems likely, and these can be difficult to find and repair.
As strongly as you are describing it to be, if what you are smelling is confirmed to be hydrogen sulfide, I would look for another apartment sooner rather than later.