Well, I would say it means to act unbothered and chill. Like-“Quick, Susie is coming, act cool so she doesn’t suspect anything”. But in the sentence you wrote, it would mean like he acts a certain way to attract the girls. Also relaxed, confident and trying to act like he’s not that fazed by anything

There's not really a specific definition, but it's a combination of confidence, charisma and casualness. One school of thought is that if you "act" cool then you probably aren't. The other school would advocate "fake it 'til you make it." The older you get, the less you care about cool and the more you focus on character.

to act cool = to act calm, sophisticated, and popular.

It's not Jim's real persona, he is modifying his behaviour to appear cool to other people. Cool is different to different cultures and eras. So something cool in Russia might not be cool in Germany, vice versa.