Please excuse my bad english. I lost a lot of knowlegde during the time after school. I hope you know what I mean...
First, let me say that I hate it when people who speak English as a second language apologize for their "bad" English. Generally, it's perfectly fine and you specifically, are perfectly comprehensible.
As I like to say: "You speak my language a heck of a lot better than I speak yours"
How is the steel? Does it rust a lot? Is the 5150 steel a carbon or a stainless steel?
Cold Steel uses 1055 for their axes, not 5160. From my experience 5160 is a bit tougher, more wear resistant, and holds a better edge than 1055.
It also has great performance over a higher range of hardness and still retains a nice level of toughness when hardened up to 58-60hrc. From what I've seen, 1055 tends to be kept around 56 hrc or below... if people want harder, they'll generally go with a higher carbon 10xx, like 1084 or 1095. For axes and such though, 1055 is a perfectly good choice as it's plenty tough and mid 50s hrc is more than sufficient for an axe. That said, my experience with cold steel's tomahawks has been pretty favorable - good impact resistance and takes/holds an edge pretty well.
5160 is also tad bit more corrosion resistant than 1055 due to it's actually having some chromium, but it's hardly a noticeable difference: 5160 has 0.7-0.9% chromium, whereas 1055 has zero (i.e 1055 is about as "non-stainless" as steel gets). 13% or more constitutes "stainless", so both are carbon steels and will rust pretty easily if not oiled and looked after. You don't want stainless steel for a heavy impact tool like an axe though - with stainless steels, you sacrifice toughness and edge retention for stain resistance, which is not a good compromise for such a tool.
How about the black-colour axehead? Does it loose that colour very fast by using?
I honestly can't comment on how well the coating stays on. One of the first things I did when I got my trail hawk was strip this coating off then force a protective patina with vinegar. I hate the look of coatings, especially when they begin to wear away...
How about using it in Heavy Duty Woodwork with chainsaw, heavy axe and hammer for shmashing a one meter wood in two. Does the hawk keep that work?
Really, I see the cold steel tomahawks as more an item for light weight hiking/camping or novelty (like throwing)... if weight isn't an issue, I can't think of any tasks that it excels at over a more traditional axe in the same length range (like a Gransfors Bruks Small Forest Axe). The shape isn't very good for splitting and the length isn't ideal for chopping. I can't think of where it would fit into your intended task, unless you need something light weight for limbing...?