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Alternative answer:

Penny

I can be used to mark your place,

Coins are often used as golf markers

When, with less than ten, you will face,

You will have to use pennies in some denomination if you have less than 10 cents

I can be found at your feet,

Find a penny, pick it up. You know the saying.

There is a flat surface at my peak,

Either a reference to the bank on the coin (the top of the bank is flat) or the edge of a coin which is flat.

I come in once the votes are cast,

Money is donated during elections, or perhaps something to do with a count?

With me you can try to visit the past,

Coins have various prints throughout the years. You could also used money to travel.

I'll demonstrate for you a melody.

Pink Floyd's "Money" has the sound of coins clinking/a cash register/the cha-ching sound of money, or perhaps a reference to when a coin spins around and around before it falls>?

An equivalent is a piece of me.

There are different denominations of coins.

TITLE:

Coins prevent physical money from being entirely note-based.

Partial answer, but enough of these are spot on that I feel like I gotta be close:

Note

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,

Footnotes

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...

My improved and new answer is:

"Pee"?

I can be used to mark your place,

I pee to mark my territory

When, with less than ten, you will face,

my range is less then ten feet

I can be found at your feet,

again I peed on the ground to make my territory

There is a flat surface at my peak,

the liquid always tries to be straight at the surface level

I come in once the votes are cast,

This ones a bit more complicated: people need to wait in line for a long time, so once every one is done vote everyone mostly goes pee

With me you can try to visit the past,

archaeologist/vets/scientist use urine samples to determine information about the animal's past health and their past environment

I'll demonstrate for you a melody.

peeing makes a distinct tinkle sound and a rhythm can be heard (different
grounds/toilets produce different melodies)

An equivalent is a piece of me.

if you cut pee in half you still have the same pee

Title: A Paper Record is What I Hamper

if you have a paper record of your pee, you probably have some disease or
health problems. So i would want to hamper a paper record of my pee(stay
healthy).

I think it's...

a ruler

I can be used to mark your place,
When, with less than ten, you will face,

You can mark you place with a ruler, some tape measures have a max of 10'.

I can be found at your feet,

Rulers measure in feet.

There is a flat surface at my peak,

The top of a ruler is flat.

I come in once the votes are cast,

A ruler ascends to power once voted in.

With me you can try to visit the past,

You can visit the past reading history based on the rulers of those times

I'll demonstrate for you a melody.

There's a band named Melody Ruler... or... melodies or songs are written in measures.

An equivalent is a piece of me.

A broken ruler is still capable of measuring lengths.

Title:

A ruler can be set on a piece of paper to hamper it when drawing lines so that it doesn't move, or simply as a paperweight.

Wildly different direction, long shot. Trying to focus on wordplay more.

A cross?

I can be used to mark your place,

Crosses are often used in graves in place of headstones

When, with less than ten, you will face,

'<'X makes a rather funny emoji/face (sorry, don't know how to escape it properly)

I can be found at your feet,

X marks the spot!

There is a flat surface at my peak,

The top of a cross is flat!

I come in once the votes are cast,

Ballots are typically marked with an x

With me you can try to visit the past,

Worldplay on "ex" - ex-girlfriend or boyfriend?

I'll demonstrate for you a melody.

Ex. is often short for example; short features on songs are often denoted with "Artist x Artist"

An equivalent is a piece of me.

Could this be a reference to mathematics, eg. 'x' is an unknown quantity and as such parts of x are still denoted as number*x?

TITLE:

This might be a reference to the 'x' on a contract/formal document, usually denoting a necessary signature to complete/officiate the document.

Are you a

Digit?

I can be used to mark your place,

Like a queue ?

When, with less than ten, you will face,

Digits are numbers 0-9, so not over 10, that uses multiple digits

I can be found at your feet,

Toes can sometimes be called digits

There is a flat surface at my peak,

Toes have toenails, which are flat

I come in once the votes are cast,

Votes have numbers, aka digits.

With me you can try to visit the past,

I'll demonstrate for you a melody.

Not really sure about these ones..

An equivalent is a piece of me.

Basic math?

Are you a

Stamp?

I can be used to mark your place,

As in place stamps to imprint your address?

When, with less than ten, you will face,

The receiver will have to bear the charges of the received letter if the postage stamp value is less than 10 cents (very cheap by the way)?

I can be found at your feet,

Foot prints?

There is a flat surface at my peak,

The stamp's peak/bottom surface is almost flat, execpt the engraved/embossed bits.

I come in once the votes are cast,

A stamp seal to seal off the envelop.

With me you can try to visit the past,

Postage stamps may reveal some history. Also, the time-stamps for digital content.

I'll demonstrate for you a melody.

Tap dancing. The music produced by stamping the feet.

An equivalent is a piece of me.

The imprint of a stamp represents the stamp's text/content.

Another answer. I think that it fits quite well with the first part and almost the second one as well.

Score

I can be used to mark your place,

Your score determines your place/position, as in baseball.

When, with less than ten, you will face,

9 (less than 10) players in a baseball team.

I can be found at your feet,

Touching the base with your feet to secure a score.

There is a flat surface at my peak,

A maximum of 4 sores at one hit.

I come in once the votes are cast,

The score after counting the votes.

With me you can try to visit the past,

A score can be an account of an event.

I'll demonstrate for you a melody.

Score of a musical composition.

An equivalent is a piece of me.

Not sure about it. May be 'core', as in essence, as an equivalent of score, or a partial musical composition record is still a score.

I think the answer is

Pole

I can be used to mark your place

You can place a flag pole to mark your place.

When, with less than ten, you will face

Could this be a reference to "Wouldn't touch you with a ten foot barge pole"?

I can be found at your feet

Is this to do with "Walking poles" used to help in fell walking?

There is a flat surface at my peak

Poles generally aren't pointed on top

I come in once the votes are cast

I think this is a reference to "The polls are in"

With me you can try to visit the past

I am still trying to figure this one out.

I'll demonstrate for you a melody

You can play a song on a flute which is usually pole shaped.

An equivalent is a piece of me.

Still no idea on this one.