I'm not actually sure what you're asking with the first question but
* You can use any email as your "Microsoft ID", it doesn't have to be Hotmail/Outlook.com. Be warned that if you somehow end up with two different IDs there is AFAIK no way to merge them and this situation is fairly common and is arrrrggghhhh
* If you want to use the Mail app with Gmail you can set this up under the settings charm.
* You can also pipe your Gmail into Hotmail/Outlook.com if you want (this is different and not required for the thing above, which just uses Gmail directly)
also
* Yes for better or worse the search charm is the standard way to search even within apps - this takes a little while to get used to. It defaults to the app you're currently in (or "Apps" if you're on the desktop or in Start) or you can switch to a different app (which lets you quickly search multiple apps with the same search terms - you can right click & pin apps you want to search frequently so they're easier to get to). Also, Start and many apps (including the store) let you search by just typing from the front page.
* Yes the Windows, Windows Phone and Xbox "marketplaces" are all totally separate and buying stuff on one gets you jack on the other ones. Why? Cause Not Invented Here, that's why.
also a tip that makes mouse usage better for me, to use the charms I find it feels a lot better when I treat it is "swing around the corner" in one motion, rather than "move to the corner and hover and move down". So Search is the upper right corner, settings is lower right