Caffeine is generally vasodilatory through the production of nitric oxide: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3003984/

That would make sense to me. Trigeminal neuralgia at least is caused by direct compression of the trigeminal nerve root, which certainly causes some degree of inflammation. Inflammation is probably a minor contributor though; demyelination seems to be the main cause of pain in that condition. NSAIDs can treat the pain and inflammation there. Anticonvulsants are more effective because they directly modulate the firing of those nerve fibers and prevent them from generating a pain signal in the first place.

I assumed that one would care about disease-related inflammation in order to treat the symptoms of the disease, but yes

I think this is a cutting edge basic science question. From what I can tell (eg this 2021 review) we are still in the early stages of understanding the link between inflammation and depression in the first place. Similar with COVID-19, though even more nascent.

In general NSAIDs have not been found to effectively treat depression or COVID-19. They have been extensively studied and I suspect we would have discovered a correlation if one existed.

More context please. Do you mean inflammation of the central nervous system (meningitis, cerebritis, ventriculitis), peripheral nervous system (neuritis), or...? Do you have a specific clinical or research concern?

NSAIDs cross the blood-brain barrier in rats. Since they are anti-inflammatory by definition, maybe yes? Again it depends what you're really looking for here -- "neuroinflammation" is a subtype of inflammation and I'm not sure the distinction is really relevant beyond the BBB, but it's not my field.

Who knew??

Control of the facial muscles comes from the 7th cranial nerve, which originates in the brainstem. This "primitive" part of the brain is divided in half but not separate hemispheres, and many signals cross the midline there.

Blinking is an easy one since the muscles are symmetric on both sides. Check out how our eyes move together for something really ridiculous.

The other responses are good. I would just add that there are a few conditions that cause fluid retention (eg congestive heart failure) which can increase the pressure in the cardiovascular system. This increased hydrostatic pressure can be measured at the heart using echocardiography. Blood vessels are generally not thick and sturdy like heart muscle, and so the increased pressure causes fluid to leak out of the vessels ("extravasate") and leads to swelling ("edema"), typically of the lower extremities due to gravity.

So there are conditions that lead to overproduction of the cellular components of blood (polycythemia, essential thrombocytosis, leukemia), and different conditions that lead to retention of the liquid component -- but the liquid just leaks out into the soft tissue.