The President can, at least in theory, pardon himself for federal crimes he or she may have committed. That does not protect that President from impeachment proceedings, however, and the conflict of interest would be earthquake-level and utterly ensure that President were run out of office in moments were it to occur.
But, as noted, this requires actual crimes to have been committed, and the DUmmies' fantasies about this don't count.
I agree. The pardon power is what is known as a 'Plenary power,' i.e. it's complete unto itself and does not require any condition or cooperation from anyone else to use. It is theoretically possible to do this, but it wouldn't protect a President from impeachment, if anything it would spur it on. A pardon relieves a person from criminal liability, which as the Nixon pardon showed, could be as broad as Cardinal Richelieu's writ from
The Three Musketeers. However, impeachment is not, strictly speaking, a criminal prosecution. There is no sentence imposed, the consequence is simply removal from office, which does not even technically prevent the removed official from running for the exact same office again in the next election cycle.
The pardon does not stand in the way of impeachment proceedings, so there would be no point in a President doing this unless he was simply interested in avoiding jail time and criminal conviction after concluding that his own removal from office was inevitable.
But it is great fun to watch the DUmmies fap themselves into a frenzy of fear and rage about the very prospect of such a ridiculous proposition.