Can’t find “People” category in Outlook.com

As far as i know, you cannot change them. See here.

You can't do that under gmail's contact quick search but you can sync your contacts and use your preferred application (or even Outlook) to search them. As a bonus, you get synced calendar.

Another option within gmail is enabling keyboard shortcuts, typing g+c then / to search, and g+i to go back to inbox. At least to me that's quick enough.