The whole concept of 'mobile' is redundant, what you need is responsive - something that adjust to all devices. And you do this using @media in your CSS not jQuery
According to W3 Schools "With a @media query, you can write different CSS code for different media types". I'm not sure that's truly "responsive" in the same sense as Bootstrap and JQuery Mobile. JQuery itself, as you rightly point out has nothing to do with mobile or responsive - it's a framework for manipulating the DOM. You can do practically everything the Jquery does in Javascript, but not sure why one would want to restrict oneself to that?
That's old information. Media types now include the screen resolution.
Using @media in your CSS you set styles for a whole range of different widths. For example, you can change the layout, menus, headers footer and everything else with just a few lines of CSS that adjusts depending on the screen resolution and if device is held portrait or landscape.
No need for any JS or jQuery anywhere. Throw in a bit of CSS3 and you can do some astounding transformations.
This is not to say jQuery doesn't have it's uses but styling a site isn't of them.
But I'm saying you don't need jQuery mobile on top of CSS3 at all. CSS3 is all you need. And after trying to use Twitter bootstrap (and others of the same ilk) I've gone back to a simpler framework. It's a bit like the responsive grids that were popular for a while, all very clever but mostly unnecessary.