Melody admirer I am, not many sense the truth. What am I?

Question

Melody admirer I am, not many sense the truth,
I belong to everyone, old and the youth.
While that little guy stands upright,
I jump atop, while gravity joins the flight.
I own a skull, but struggle to think,
Have an amazing eye, but struggle to wink.
I do have a talon, and yes it plucks,
Having a neck with no food sucks.
Dare you try provoking my head,
Every now and then I fancy the bloodshed.
I ain't precious though being owned by a lord,
Modest I am, anyone can afford.
I've spent ages with the white stones to thrive,
Signs of love were embossed then and for eternity they will survive.
Listen carefully, I need to warn you,
I arrive with M plus V-squared by two.
Paired with one of my type I believe in symbolism,
Together we appreciate the idea of socialism.
Often during criticism and excitement I am referred,
Let me give you a hint, I am a six letter word.

Answer explaining each line correctly will be accepted!

So go ahead and tell me - What am I? - Enjoy :)

Answer 1

Clearly (after recent edits) you are a

HAMMER.

I don't think I've understood everything in the riddle quite as intended, but here's what I have:

Melody admirer I am, not many sense the truth,
I belong to everyone, old and the youth.

"Melody admirer" might be because some musical instruments are played using hammers. (The piano has lots of little hammers, actuated by the keys. A dulcimer is played by actually hitting the strings with hammers held by the player.) I think "not many sense the truth" is just filler, but M Oehm (in comments) makes the suggestion that it might refer to the fact that one of the bones in the middle ear's hearing mechanism is called the malleus or hammer; this of course would also fit with being a melody-admirer. Does the next line mean anything more than that lots of people have hammers? (If we're talking about the one in the ear, then indeed pretty much everyone has them.)

While that little guy stands upright,
I jump atop, while gravity joins the flight.

[EDITED to replace an unnecessarily complicated explanation with a much simpler one offered in TSL by Sconibulus:] When hammering in a nail, often the nail (the "little guy") is upright, the hammer moves up and down above it, and gravity helps.

I own a skull, but struggle to think,
Have an amazing eye, but struggle to wink.

There appears to be a weapon in some video game called a "Skull Hammer", which might possibly explain the skull (though I'd have thought a Skull Hammer should be a hammer for smashing skulls). Otherwise I don't understand the skull. The eye is the hole in the (usually metal) head portion through which the (usually wooden) handle shaft passes. EDITED to add: OP explained in TSL chat that "I own a skull" because hammers have heads and heads are kinda like skulls :-).

I do have a talon, and yes it plucks,
Having a neck with no food sucks.

A "claw hammer" is one with one side of its head bifurcated for levering nails and things out of whatever they're stuck into. On the other side there is generally a more conventional cylindrical head, with a narrower part called the neck.

Dare you try provoking my head,
Every now and then I fancy the bloodshed.

Hammers have been used as weapons of war. (And it's the head of the hammer that you might get hit with.)

I ain't precious though being owned by a lord,
Modest I am, anyone can afford.

Perhaps the lord is the god Thor? In any case, hammers are generally not expensive.

I've spent ages with the white stones to thrive,
Signs of love were embossed then and for eternity they will survive.

I guess this is about diamonds, which I assume are mined using hammers. (They are white stones, near enough, diamond jewelry is traditionally given to one's romantic partner, and "diamonds are for ever" as the de Beer family has hornswoggled us into believing.)

EDITED to add: actually this was wrong and the OP had in mind

the Taj Mahal, made from white stones which were presumably quarried and carved with hammer and chisel.

Listen carefully, I need to warn you,
I arrive with M plus V-squared by two.

I guess we're back to the hammer as weapon, this time for throwing. The damage it does is a result of the kinetic energy ($mv^2$) the hammer has.

Paired with one of my type I believe in symbolism,
Together we appreciate the idea of socialism.

Hammer and sickle.

Often during criticism and excitement I am referred,

Hammer and tongs. And you might describe really severe criticism as a "hammering".

Let me give you a hint, I am a six letter word.

H, one. A, two. M, three. M, four. E, five. R, six. It works! :-)

Answer 2

This is a shot in the dark, but I think you are...

An arrow. (Yes, "shot in the dark" is a pun).

Melody admirer I am, not many sense the truth,

An arrow is launched from a bow. In a way, it is a "stringed instrument," although most people would not consider it so.

I belong to everyone, old and the youth.

Arrows were used by ancient civilizations, and are still used today for hunting and sport.

While that little guy stands upright,

Usually, the archer holding the bow is standing.

I jump atop, while gravity joins the flight.

Arrows are launched in upward trajectories, and gravity makes them arc downward.

I own a skull, but struggle to think,

Some arrowheads are made of bone. A "head" made of bone is a skull.

Have an amazing eye, but struggle to wink.

This is a creative one. It uses "have" in the sense that the arrow "has" an archer who launches it. The archer needs good aim, and winking means they can't see as well.

I do have a talon, and yes it plucks,

Another creative use of "have". The archer fires the bow with their finger ("talon") to pull and release the bowstring. This is like plucking a string.

Having a neck with no food sucks.

The shaft of an arrow is like a neck, because it connects to the arrow "head".

Dare you try provoking my head,

The arrowhead is typically sharp, and is what causes injury.

Every now and then I fancy the bloodshed.

Arrows were often used in wars, for ranged combat.

I ain't precious though being owned by a lord,

Individual arrows are cheap. However, a lord's army of archers was incredibly powerful, especially in ancient warfare.

Modest I am, anyone can afford.

Again, individual arrows are cheap. You can make an arrow from a thin strip of wood, plus a rock or sharp bone.

I've spent ages with the white stones to thrive,

Multiple possible meanings. Bones and rocks may be used as arrowheads. Or, the dead who are killed by arrows leave behind bones.

Signs of love were embossed then and for eternity they will survive.

Arrowheads crafted from stone required lots of time and focus for carving.

Listen carefully, I need to warn you,

If you have very keen hearing, maybe you can hear an arrow in motion.

I arrive with M plus V-squared by two.

I'm interpreting this as (1/2)mv^2, the formula for kinetic energy. The reason an arrow projectile is so deadly is that it has kinetic energy carrying it forward.