For example, if President Trump nominated a justice on his last day in office, could that person then be confirmed by the Senate during a Biden administration?
Yes and no. It's theoretically possible for the Senate to confirm a candidate nominated by a previous President. It's important to remember that even though it's commonly called "confirmation," the word used in the Constitution is "consent." So the Senate merely consents to the appointment of a candidate, and the President can still decide after the confirmation process not to follow through with making that appointment.
(Notably the Constitution doesn't say anything about "nomination," so the Senate can "confirm" anyone they wanted. The nomination process only exists as a practical measure because the Senate does not want to waste their time going through a lengthy and intensive confirmation process for candidates that the President isn't going to appoint.)
But this couldn't be done as a way for the Senate to force through a candidate that the sitting President doesn't like. Theoretically, Joe Biden could still decide to appoint a Trump nominee, but he would not be required to, and given the current political climate in the US, it seems unlikely that he would.
(It's also worth noting that for the same reasons of practicality mentioned above, the President can withdraw a nomination at any point in this process. Wikipedia has a list of Unsuccessful nominations to the Supreme Court of the United States, which lists 11 cases of withdrawn nominations.)
Of course this is all speculative, since it hasn't happened, and a President who wished to appoint a candidate nominated by a previous administration would most likely officially re-nominate the candidate and let the Senate re-confirm. (In your example of the last day in office, or even going back a couple weeks, the Senate that confirmed the original nomination would still be the sitting Senate, since the new Congress is seated on January 3, while the President isn't replaced until January 20. Barring any major developments, there would be no reason to expect the Senate not to re-confirm.)