Yes, it is for members only.... =)
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No. Skill capes are members only.
Yes. All skillcapes are members-only.
That's debatable, but I'd say cooking if you have the money. Fletching is another fast one.
The cooking skill is just like in the base game. You get the cooking skill from cooking.
Yes, you have to be a member to obtain ANY skill capes in Runescape. Skill capes were intended to be made for recognizing members who reached level 99 in a skill, not really for non-members. Although, a common trick to obtain the skill cape while not cracking your budget, is to get a 1 month subscription to runescape, buy the skill cape, equip it to your character, and never take it off again. If you take it off, you can't put it back on once your membership is depleted. Although, this was a relatively old trick and I do not know if this still works.