I am making the assumption that this is an iPhone app and by "full-width popup" you mean "Modal View Controller".
Use the default "UISearchBar" - it has a cancel and a clear button. Users will use the keypad to confirm the search (if the search isn't live). And "Cancel" will close the modal. : )
While I think the cancel/back button is helpful, I'm not sure that it's the most effective. Will the page change on cancel/back, or does it just hide the search interface? Is there any function that clears the search box?
As far as the placement of the search button, I would definitely recommend adding it, unless you're certain that your typical user is used to the pattern of hitting enter to submit.
From UX Movement:
Having a search button (or link, in your case):
- describes the action the system will do
- makes the button easier to click
- does not require any additional labels
- is simple and clean
designing an app UI you should
"Keep It Simple".
back button is crucial to your app than you can go with
option 2 but if not than go with
option 1 because it represents a search bar with all the required functionality that a user needs. You should not over complicate things for your users.