The notes that a piece of music are based around are called the motif. And it can differ based on how a composer wants to use it.
It can be something the composer wants to be familiar so will constantly return to it, such as with an iconic film piece like Raiders of the Lost Ark. Or it can be a short sequence of notes that a composer plays with throughout the piece (such as Beethoven's Fifth, where you have the four notes, and they are constant throughout but used slightly differently as the piece goes on.
And yes, really, there is no such thing as originality in music, when you really boil it down.
Bono, of U2, once said "all poets are thieves", and that's really true.
There is no sequence of notes or chord progressions that have not been used many times before, which is one major reason why I think copyright laws are bad ideas. Not only do they stifle creativity (because they forbid you from using anything that is "too close" to another piece or song, but they treat every individual song as if it's an original work, which in one sense it is because you are writing something where the individual words, notes, chords, etc., may not have been used specifically in that way before, but in a more real, important sense you are not, because no one owns a chord progression or the notes you are using and in what order you are using them.
So, for example, when Katy Perry got sued for her song Dark Horse which used a bass progression that was similar to Marcus Gray's Joyful Noise, though it was in a different key and in a different sonic setting, it was arguably a bad case because while the motif is similar, it's still just a series of eight notes, dotted eighths, and others, with a very limited musical vocabulary (it's not like in writing, where there are thousands upon thousands of words you can use).
So not only are copyright laws pernicious, they can be used in unscrupulous ways for artists to get money that they don't deserve from unsuspecting artists who meant no harm.
I don't think it's possible to really put a number on how many pieces of music use short motifs like that. They'd be more numerous in classical, most likely, than modern pop music because pop music focuses on catchy melodies that get songs stuck in your brain and make you want to buy them.
Classical pieces weren't as concerned with that because sound recording technology had not come about yet. And anyway, classical composers were less concerned with plagiarism (variations on another composer's pieces were fairly common back then).
The attitude among composers was generally one of feeling honoured that another composer would use their melody in a composition, as opposed to now where the attitude is generally "you better not copy any element of my song or I'll sue you".