Partial answer, but enough of these are spot on that I feel like I gotta be close:
I can be used to mark your place,
You can mark your place with a note
When, with less than ten, you will face,
I'm fairly certain this refers to the spaces on a musical staff which represent the notes F, A, C, E, but this only works on the treble clef, which is why I'm confused by "less than ten", which I read as "below tenor", but the only common clef below tenor would be the bass clef, where the spaces read A, C, E, G, unless you mean the spaces below the lines, in which case you've got F, A, C, E, and um, G, but maybe we ignore that?
I can be found at your feet,
There is a flat surface at my peak,
Barred notes like ♫ often have a flat surface on top.
I come in once the votes are cast,
This one I'm not sure about. Maybe notes from news pundits reporting on elections or something?
With me you can try to visit the past,
You can take notes to try and remember stuff later.
I'll demonstrate for you a melody.
Musical notes, clearly, as discussed above.
An equivalent is a piece of me.
Another word for note is "memo", and a piece of "memo" is "me", though that's kind of in reverse, but wordplay is like that sometimes...
A Paper Record is What I Hamper
Notes are of course, paper records. Possible dig at a certain head of state recently complaining about lawyers taking notes hampering his ability to lie about stuff later? Trying to keep the politics out of the puzzling...