On tea tree oil, I know of one case of one doctor trying it internally as a test. It induced gastro-intestinal problems.
Do you happen to know what kind of doses are used on those experimental animals to create an inflammatory state? Reason I ask, is because from the review I posted above, it says it is anti-inflammatory.
The paper you cited — which I found interesting to read — also says that alpha-pinenes were found to inhibit AChE (= acetylcholinesterase), which is interesting, because acetylcholinesterase inhibitors seem to be of benefit for ME/CFS, especially for sleep (see this study).
Certainly I don't think you are going drop dead if you take small doses (a few drops) of turpentine for a few weeks, provided you treat it as an essential oil, and always dilute these drops in some cooking oil before swallowing. You are normally advised to dilute your few drops of essential oil in 15 to 30 ml (= 1 to 2 tablespoons) of ordinary cooking oil. Essential oils should never be taken neat.
The main alpha- and beta-pinene ingredients of turpentine are also found in reasonable amounts in rosemary essential oil, so you might want to consider taking rosemary essential oil instead, as a safer version of turpentine. Around 22% of rosemary essential oil is alpha-pinene.
Juniper berry essential oil is even better, as this contains around 44% alpha-pinene (ref: here).
So I think either rosemary or juniper berry essential oils are a much better alternative to turpentine.
Rosemary and Juniper berry essential oils are listed in the FDA's list of essential oils that are generally regarded as safe (GRAS), so you can take these internally, again in dosages that are measured in drops.
Alpha-pinene is found in many other essential oils, as you can see here, but I think juniper and rosemary may have some of the highest percentages (unless you can find another essential oil that contains more).
You can buy essential oils quite cheaply on eBay: see here.
According to the paper you cited, the greatest danger in turpentine is the carenes ingredient. This study details some of the toxic effects of carenes. I think the CNS symptoms are the most concerning; I'd prefer not to ingest toxins that affect my brain!