What makes a cool video game protagonist?
Super Puppy! That would be very likable.
Seriously, though, emark. What kind of answer are you looking for? You named some protagonists but you didn't say what makes them "good and likable," or what you mean by "what makes a very good and likeable protagonist." Are you looking for analysis? If so, give us your analysis. Or are you just looking to make a list? And if so, is it a list of existing game protagonists or a list of ideas for protagonists?
Everybody has different preferences that draw them to a character, whatever it may be. I think its mostly due to the image the character presents in the aesthetic and overall setting of the game. I tend to be drawn to characters who aren't so romanticized. I like A superpower here and there is cool, but I like characters who are average joes caught in a rut as opposed to some super soldier sent to fight off a race of man eating monsters.
The answer to your question is very subjective. Using Master Chief as the blandest example and a base, I'd say the elements you seek in a likable protagonist are:
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Dry one-liners, prowess as an action hero, detailed backstory and camaraderie with others.
Whilst not directly pertaining to a "cool" character I have found the following articles to be of use at times
In my opinion a cool character is one that shines in the situation they're given. Just think, would Master Chief or Sonic be just as cool if they were in a game like Portal or something like Pokemon. Having the setting and protagonist complement each other is the easiest way to create a bad ass character. If you already have a setting in mind then work on your character based off that or vice-versa if you've decided on a character.
Also (like -Crimsix- said) "cool" is subjective, many characters are cool because they're confident in their abilities.
The best protagonist is one that is 'real'. I don't mean they can't do things like magic, or can't spawn random tentacles on a whim, or pop brains from afar. Rather, a real protagonist is one whose motivations are clear, and if I were to know what they're motivation for doing something is, then their story would make sense -- even if it was batshot idiotic.
I'll use an anime as an example (forgive me for my niches). In GunXSword, Van the protagonist accompanies a girl, Wendy, on her journey to find her brother who has gone missing. Van could care less, as his motivation is to find the man who killed his wife. Thus, knowing his motivation, everything that he does relating to this plot should be caused because he is looking to find the man who killed his wife. For example, The only reason Van allows Wendy to accompany him is because they are going the same way. The only reason Van meets the people he does is because he is looking for this man. If he wasn't, none of this would be happening, and that makes him real.
A bad protagonist would be someone who 'saves the world just because'. Someone who goes on an incredible journey just because someone told them to, while their motivation would be something akin to 'chilling at home and eating pizza'. These protagonists don't feel real and there are so many of them, but the good ones are the ones we remember, like Lara Croft, Solid Snake, even Link has motivations -- his only wants to protect Zelda. Before that, he's basically nobody.
So, to sum up, the real protagonists are the best protagonists, and the best protagonists are the coolest. Hell, even Ocelot is cool and he's an antagonist, but his goals are clear and he feels real and someone, somewhere in the world, should relate to him.
