What happens when you use a name tag on a strange weapon

To answer your title's question, strange flamethrowers only count kills done by that weapon's fire damage. This is either through direct damage (W + M1) or the afterburn. It does NOT count reflects sadly. Taken straight from the TF2 Wiki's strange page:

The following actions do not increase the total kill count:

• Killing an enemy with a compression blast.

Strange - TF2 Wiki

As for kill counts, I have a strange degreaser and flare gun too and my flare gun does have a lot more than the degreaser. My guess is in how I play. I tend to ignite, air blast and follow up with a flare gun shot or the axtinguisher. I reflect a lot also (but those kills don't count). If at range I use the flare gun for ignitions and annoying. Since the afterburn damage on the degreaser is less, then you are less likely to get a kill from that than the other flamethowers. Also if you do the air blast and flare gun, the flare gun's afterburn will override the degreaser's so if you get the kill after that, it's for the flare gun.

I would so love a separate counter on my degreaser for reflect kills, like the sentry kill count on the engy's wrench.

It will only count when you "dominate" an enemy for the first time. When you hear that familiar sound that indicates a domination, it should increase your counter by 1.

Subsequent kills on a dominated enemy will not increase the counter for the part. However, that same player can remove the domination by:

• switching teams, then switching back
• leaving the server, then rejoining
• getting revenge

If any of these three things happen, the previous "domination" will be wiped, allowing you to dominate the player again, if you wish.