I don't see authenticity as one of your criteria... It appears you are looking for a purely functional tool. You have a lot of options.
I do see a possible conflict, in #2 and #5. By "firesteel," I'm assuming you mean a ferro rod. If you are thinking of striking an actual flint against your axe, your axe would
be the firesteel. Some clarification would help. Anyway, you need pretty hard steel to scrape a ferro rod, and the harder the steel, the harder it is to sharpen. I like to be able to sharpen my axes with a file. This puts a good woodcutting edge on it. If I want to shave, I use a razor.
Modern (late 19th - early 20th century) axe design evolved for a reason. In my opinion, based on personal experience with various axes, hatchets, and tomahawks, classic early modern axes simply work better than tomahawks for camp chores. However, I like tomahawks for the same reason I would rather shoot a muzzleloader than a .30-30.
"Throwing Hawks" are a late 20th century phenomenon. People on the frontier absolutely threw axes and tomahawks; this is well documented. However, it was some time in the mid-20th century when the NMLRA codified the rules for competitive tomahawk throwing, and the "throwing hawk" was developed to stay within the codified guidelines while still pushing the limits of what is allowed. If you look at an old-time trade axe, like this one...
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... you'll see that the topline of the blade is level, and the bottom line runs straight. The width of the cutting edge is variable. In contrast, the topline of a "throwing hawk" flares upward, to maximize the width of the cutting edge (I think the rules allow 4"), and the bottom line is curved to take off weight. Also look at the handle. The trade axe has a sturdy handle, and this one actually has a knob carved on the end, to facilitate a better grip. On the other hand, a throwing hawk handle is tapered down to what I consider an uncomfortably small diameter to enable a clean release in throwing at a target. I would suggest that for your purposes, it would be best to steer clear of throwing hawks.
For driving stakes, you'll want a flat or hammer poll. This picture shows a Shrew Camp Axe that I bought from the late Ron LaClair some years ago.
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The handle is straight, not curved. The appearance of a curve is just a photographic artifact. Anyway, it does have a nice, beefy handle so you can get a good grip. The head weighs about a pound, and the total length of mine is 16-1/2", with a 2-7/8" cutting edge. You can see it has a "hammer poll." I have not tried it on a ferro rod, nor have I tried knocking sparks off of it with a flint, so I can't comment on that. I think Ron designed these and marketed them from his archery shop, but they were actually made by H&B Forge, and the specs on this axe are very close to the
H&B forge Medium Camp Hawk. These cost right at $100 now, but I didn't pay near that much for mine. They have really gone up. However, this is a genuine, high-quality hand-forged axe, and it looks exactly like an original from New Mexico that I saw pictured in a book I have. It doesn't chop or split wood as well as a more modern design, but it works pretty well for all of that, and it is a mighty handy little tool, while still being traditional. I would recommend it.
If you are less concerned about authenticity or tradition, I might also recommend one of these:
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These little axes are both called "Flying Foxes." I don't know if these are legal under NMLRA tomahawk-throwing rules or not. The heads on these two are identical, weighing about 1-3/4 pounds, with a cutting edge just a hair under 4". The only difference is the type of handle. For general use, I prefer the straight handle, but that's just personal preference. The axe with the curved handle is about 15-1/2" long, while the straight-handled one is about 20-3/4" in total length. I have not used either of these a lot, but I have used them some. I got mine from
Whiskey River Art and Trading. I bought the straight one when they had a sale, and then a little later they had some kind of deal that included a free Flying Fox. I requested the curved handle model for my freebie. These are made by Council Tools and are priced under $60.
Strangely enough, the curved-handle Flying Fox was designed for throwing in modern-day axe-throwing competition. I understand there are pubs or venues in a lot of cities where people can go to compete or just try it out. I've never tried it. You get a lot of young adults together in a confined space, drinking beer and throwing axes... What could possibly go wrong?
Best regards,
Notchy Bob