This is an interesting question. You are certainly not an isolated case.
The "origin" of foot odor has been examined in many studies.
This study conducted by Marshall et al focused on the microflora
found on normal feet (no foot odor) and compared it to feet with foot
odor. They reported a higher population densities of staphylococci and
aerobic coryneform ...
Certainly, headaches can be caused by a variety of smells, perfumes, paints, gasoline, bleach and yes, air fresheners.
A study in 200 migraine patients showed perfumes were the first most common cause when headaches are attributed to odors (75%), followed by paints (42%), gasoline (28%) and bleach (27%)
I wouldn't be surprised if dogs could detect COVID-19, as they can help detect hypoglycaemia in diabetics, warn people who are about to have an epileptic seizure and have been used to sniff out some cancers.
This article on Nature's website (23 November 2020) has the following
Canines seem to detect coronavirus infections with remarkable
accuracy, but ...
According to the summary of one study:
Of 2,581 COVID-19 patients studied, 95 percent of patients regained their sense of smell within six months, according to the study in the Journal of Internal Medicine.
The study appears to be this one (on patients from continental Europe) and it has more details even its abstract:
A total of 328 patients (24.1%) did ...
The first question I would ask in response to your question on gum disease is "Have you spoken with your dentist about the health of your gums?"
Looking at the NHS webpage on Halitosis, it sounds that most of the possibilities have been ruled out, but not oral hygiene completely.
The most common cause of bad breath is poor oral hygiene. Bacteria that ...
I agree with the answers on cleaning and the type of bottle/cork/lid.
Another thing you should consider is that residue from your saliva may be what smells bad. Try leaving the freshly-cleaned bottle filled with your usual water out for a day without drinking from it and see if there is any odor, in which case it might be your water source. Then try ...
Plastic bottles work just fine. When buying sparkling water (I'm German), they come in plastic bottles and never get this taste of old shoe soles, and I haven't had any problems with bacteria either.
In your case, the cork could be a problem, as was pointed out before. I personally have made the experience that water in steel bottles always tastes a bit ...