This is really two different questions...
Second question first: I'm pretty sure that it is expressed in later that all monoliths are one and the same, just different representations in our space. Although, that then doesn't fit with the lag of 950 years for communication.
First question: the monolith on Europa in 2061 is evidently a long-lasting feature, providing some kind of shelter for the Europan lifeforms, and also, apparently, guiding them (possibly, ever since the events of 2010). I figured this would be the same for the monolith in 2001 - seems I was wrong.
Thumbing through the book I find the passage:
As he led the tribe down to the river in the dim light of dawn, Moon-Watcher paused uncertainly at a familiar spot. Something, he knew, was missing, but what it was he could not remember. He wasted no more mental effort on the problem, for this morning he had more important matters on his mind.
Like thunder and lightning and clouds and eclipses the great block of crystal had departed as mysteriously as it had come. Having vanished into the non-existant past, it never troubled Moon-Watcher's thoughts again.
A little bit later in the book, there is a reference many such 'crystals' and they seem to appear and disappear at the same time. This would indicate that the monolith(s) disappeared after only hanging around for a few nights.
However, 3001: Final Odyssey describes the discovery of a monolith in Olduvai Gorge (the site of the famous discovery of Lucy by the Leakeys) somewhere around the 26th century. It became labelled TMA-ZERO (since the lunar monolith was labelled Tycho Magnetic Anomaly One). There's even this passage:
For here was an archaeologist's treasure-trove - crudely fashioned flint tools, countless bones - some animal, some human - and almost all arranged in careful patterns. For centuries - no, millennia - these pitiful gifts had been brought here, by creatures with only the first glimmer of intelligence, as tribute to a marvel beyond their understanding.
So - depending on if we are restricting ourselves to the first book/film only - the monolith was only around for a few nights, and affected only one small tribe of proto-homonids (although, other monoliths were also doing this elsewhere at the same time). If we accept ret-cons, then it will still be around after hundreds of thousands of years, although possibly inactive once buried after an unspecified period of activity.