Unknown, but almost certainly yes
There are two impeachment incidents which seem relevant, both concerning an impeachment trial that occurred after the impeached official had left their position or been removed from it through other means.
The impeachment of Senator Blount occurred all the way back in 1797. This was in fact the first federal impeachment ever. Senator Blount was implicated in a landgrab conspiracy. The House began impeachment proceedings against him as a result. The Senate, however, had ideas of its own. They did not seem to like the idea of the House having a say in getting rid of a member of the Senate. Instead the Senate quickly exercised its constitutional power to discipline and expel a member, doing so almost unanimously.
But this did not deter the House, who continued their impeachment hearings. Ultimately they voted to impeach, and sent it off to the Senate for it to be tried. Blount's attorneys (Blount himself refused to show, and was by this point a state Senator) argued the matter should be dismissed for two reasons: mootness because Blount was no longer a US Senator, and lack of jurisdiction because a Senator is not an "officer of the United States" as in the Constitution. The Senate first voted to declare that a Senator is, in fact, an officer of the US, and so subject to impeachment. It failed to pass. They then voted to dismiss the impeachment. That succeeded. But, for some reason, they did not specify why they dismissed. Did they accept the lack of jurisdiction? Or the mootness? Or both? Or something else?
The failure to pass the resolution that they had jurisdiction, that Senators were "officers of the United States", would seem to indicate that the dismissal was in fact for lack of jurisdiction. Most believe the case is precedent establishing lack of jurisdiction, at least. But ultimately we just don't know if "mootness" was involved or not.
In 1876, Belknap was President Grant's war secretary, known for living a lavish lifestyle that seemed well beyond his government salary. Eventually it was discovered that he had been engaged in graft in a display of corruption that was brazen even relative to Grant's scandal-plagued administration.
Just before the House was to vote on articles of impeachment, Belknap tearfully handed in his resignation to Grant. This did not deter the House, who thought it would be a serious subversion of government and justice to simply let someone off the hook because they bailed on the office. The Senate had to address whether or not they retained jurisdiction and could try an impeachment of someone who was no longer in the office. They voted that, in fact, they did, and the trial proceeded. Ultimately a majority voted to convict on all counts, but they all fell short of the two-thirds mark required by the Constitution.
Thus Belknap was acquitted in the Senate, but nevertheless the trial sets clear precedent that an impeachment and subsequent trial can both occur after the accused official has left office.
From that we can conclude with near certainty that a President can similarly be impeached and tried after he has left office. But as it's never been tried—there was an opportunity to do so with Nixon, but ultimately the House did not proceed further once he resigned—, and these precedents are over 130 years old, it's a little hard to say if a modern day Senate might see things differently. Perhaps they would argue that Presidents are different with regards to impeachment somehow, or that "left because their term ended" is meaningfully different from "left by other means"; one certainly seems a much clearer active effort to avoid justice and consequences.
As for the punishment(s) that could be handed out, there are only two outcomes permitted by the constitution: removal from office, and being banned from holding any federal office or position of trust. The Senate has in fact established precedent that these can be voted on in sequence: conviction on an article of impeachment automatically triggers removal from office, and then if convicted a second vote is held regarding whether or not to impose the ban. So in your situation where the officer has already left or been removed from the post, the only punishment that could be imposed by the Senate is to be banned from future office. The constitution still permits federal and state authorities, as relevant, to pursue criminal charges on any actions that were part of the impeachment, but these would not fall to the Senate (or Congress in general) to deal with.