Tried to use this to destroy a construct. Did not go well. 

Just curious, as a Sorc with twin spell meta magic, could I twin cast this and dispel magic on 2 target creatures? Dispel barkskin on a druid and haste on a fighter for example? Also I have a second question it says ANY spell of third level or lower on the target ends. Does this mean that all of them would? If a fighter had haste, barkskin and magic weapon cast on them by 3 different casters, would dispel magic remove all of them or one of them?

Quote from BadhandsBob >>

Just curious, as a Sorc with twin spell meta magic, could I twin cast this and dispel magic on 2 target creatures? Dispel barkskin on a druid and haste on a fighter for example? Also I have a second question it says ANY spell of third level or lower on the target ends. Does this mean that all of them would? If a fighter had haste, barkskin and magic weapon cast on them by 3 different casters, would dispel magic remove all of them or one of them?

 
The way I read all of these rules. I would say: 

1. Yes, you could Twin Cast dispel magic on two targets and wipe out Haste on a fighter and Barkskin on a druid. It's basically like casting the spell twice.
2. Uh... At first, I thought No on this one because it listed one magical effect but I think buff spells are a different beast. So I think I would have to say yes on this too. If you had a fighter had Haste, Barkskin and Magical weapon on them simultaneously, then hit them with a Dispel magic, that would end all of those effects with no roll. Jeezy Chreezy, that's pretty powerful.

This might inaccurate, but I think the magical effect stuff is more for environmental or natural effects on monsters, like maybe a Beholders Antimagic Cone (that would require a roll) or maybe a magic trap on a door or something. Effects that just are, I guess is how I'm interpreting that.  

Quote from BadhandsBob >>

Just curious, as a Sorc with twin spell meta magic, could I twin cast this and dispel magic on 2 target creatures? Dispel barkskin on a druid and haste on a fighter for example? Also I have a second question it says ANY spell of third level or lower on the target ends. Does this mean that all of them would? If a fighter had haste, barkskin and magic weapon cast on them by 3 different casters, would dispel magic remove all of them or one of them?

The spell can indeed target two people with twinned as long as you burn the 3points to cast it. It will automatically remove all 3rd level effects from both creature, and you must roll a DC for each spell above 4th. Sadly it does not say you can choose to ignore a check...so if you go to remove the negative effect on a teammate it might just remove their buffs too(3rd levels are auto-removed). 

It says choose a creature, object, or effect. So as far as I can tell you can either target a creature and remove all effects, good and bad, or remove a single effect from that creature. 

From what I have read in notes and errata a single effect is "self standing". Magical barriers etc.

This spell is the reason more villains don't abuse True Polymorph.

Also can this spell work on the spell Flesh to Stone? Usually you need Greater Restoration in order to cure petrification; but can this spell, assuming you roll high enough, do the trick?

Can you dispel a magic item? Potion of healing? +1 Shield?

According to wizards of the coast, they say that dispel magic is only to break other spells. It has no effect on spell like abilities i.e. vampires charm. It also does nothing to the properties on a magical item. 

Yes, as written it's all spells on the target, making Dispel Magic perhaps the most underappreciated incredibly powerful spell in the game.

Agreed, but not only for dispelling buffs on creatures as it holds great utility in dispelling rituals which gives you a solid chance to end spells that could be absurdly powerful if completed. 

For those who don't know: this cannot be used to dispel magical effects granted by things with the duration of instantaneous. This was verified by Jeremy Crawford years ago. It's also in the PHB.

Instantaneous

Many spells are instantaneous. The spell harms, heals, creates, or alters a creature or an object in a way that can't be dispelled, because its magic exists only for an instant.

An interesting discussion on Dispel Magic for Greater invisibility here. 

Could this spell work on a permanent version of major image?

Quote from Lazuli1925 >>

Could this spell work on a permanent version of major image?

 Major Image states "At Higher Levels. When you cast this spell using a spell slot of 6th level or higher, the spell lasts until dispelled, without requiring your concentration." 

This explicitly states that it can be dispelled, which includes Dispel Magic. Most permanent effects (all if I'm not mistaken) have a duration of either Instantaneous, in which case they can't be dispelled, or Until Dispelled, such as the upcasting of Major Image or True Polymorph after an hour of concentration.

Quote from Ceada_Michael >>

This spell is the reason more villains don't abuse True Polymorph.

Also can this spell work on the spell Flesh to Stone? Usually you need Greater Restoration in order to cure petrification; but can this spell, assuming you roll high enough, do the trick?

You can dispel the effect if you cast it before the petrification becomes permanent.

What level can a sorcerer learn this spell?

(Not lvl 1)

Sorcerers get 3rd level spells at level 5. You can see these details if you navigate to the sorcerer class page and scroll down to "class features."

Casting word is:

Negalus

Does having proficiency with arcana help the DC?

It does not, from the source I looked into.