Learn English – “I’ve done everything I can” vs “I’ve done everything I can do”

The teacher is wrong. The phrasal verb is 'to give in'. You can see this by consulting any dictionary, e.g.

give in

to ​finally ​agree to something that someone ​wants after first ​refusing: If you want them to give in you'll have to ​offer them more than that.

› to ​accept that you have been defeated and ​stop arguing or fighting: After months of ​resisting the ​takeover, the ​company was ​forced by its ​shareholders to give in.

http://dictionary.cambridge.org/dictionary/english/give-in

As you can see from those dictionary examples, the word 'to' does not inevitably follow the verb.

It happens that there is a (British) verb 'to give into' but it is unusual and has a completely different meaning. It would certainly not be used with respect to temptation.

As you have discovered, one and done became popularized after a controversial National Basketball Association draft rule introduced in 2006. Recruits needed to be 19 years old by the end of the draft year and one year removed from their high school graduation before they were eligible to participate. The effect was that high school players who might have gone directly into professional play instead spent a single year playing at the collegiate level; they were not interested in a collegiate career, but would be one [year of college play] and [then] done [with college play].

But the basketball/sports recruiting meaning is not the only way to use the phrase one and done, especially outside of sports and in more casual registers. The phrase is short and rhymes pleasantly, and its literal meaning often applies: something is done once, and then done no more.

Disposable products (e.g. tissues, contact lenses, diapers, dinnerware) might be referred to as one and done referring to their lifecycle and usage pattern. There is a brand of baby wipe so named because, presumably, a diaper change requires only a single wipe, as opposed to competitors' multi-wipe offerings. Other products use it to refer to ease of use: a single application, a single transaction.

Elsewhere in sports, it has been used to refer to false start rules (to be disqualified after a single false start) and to tournament performance (to be eliminated after one round). A couple might be one and done if they plan to have only one child. In the U.S. military, a one and done is a serviceman who signs up for only a single enlistment. I can think of at least three or four ways to interpret it when referring to romantic relationships and sexual congress.

Stanley Fischer's comments can be taken to mean that if the Federal Reserve raises interest rates, it may raise them again not long thereafter. A one and done action by the Fed, in contrast, would indicate that a single rate increase would not be followed up with another, at least till the end of the year or some other milestone.