In general: You chose which AC to use

The answer to your question is very straight forward:

If you have multiple features that give you different ways to calculate your AC, you choose which one to use. (PHB)

Monk/Druid - Can use either

For Monks specifically there is an official ruling by Jeremy Crawford:

The druid/monk could use Unarmored Defense or the beast's AC calculation.

Barbarian/Druid - Can use either

Barbarian unarmored defense is worded exactly the same as the monk's so the previous ruling is applicable here as well.

Draconic Resilience - Beast's AC except, maybe, if the beast has scales

You can't use features that the beast form doesn't have

Per the PHB

can use them if the new form is physically capable of doing so

and Jeremy Crawford

Wild Shape: keep your racial traits, unless told otherwise or they require external features (wings/hands/etc.) the new form lacks.

You cannot use any racial features that your beast form does not have.

Draconic resilience depends on specific physical features

The Draconic Resilience feature says:

parts of your skin are covered by a thin sheen of dragon-like scales. When you aren’t wearing armor, your AC equals 13 + your Dexterity modifier. (PHB)

Note that this effect is predicated on the presence of dragon-like scales on your skin specifically. This means that this feature requires those to be present.

If you wild shape into a beast with skin and fur, the best thus lacks the necessary physical traits to make use of the draconic resilience.

Very similar official ruling

Jeremy Crawford made a ruling about a lizardfolk's natural armor feature. The description of which is:

You have tough, scaly skin. When you aren't wearing armor, your AC is 13 + your Dexterity modifier. (VGtM)

Note how it is very similar to the description of draconic resilience.

His ruling was:

The lizardfolk's Natural Armor specifies anatomy: tough, scaly skin. It doesn't apply if you're wearing a beast's skin in Wild Shape.

However, if the beast has scales it might qualify (DM's choice)

Note that in the above ruling Jeremy Crawford specifically says "wearing a beast's skin". It is possible that if the beast has scales (for example the crocodile or ankylosaurus) that it might qualify as now having the necessary physical feature (scales) thus allowing you to use draconic resilience.

This will likely come down to a DM call as to whether any of the creatures on the list have a "thin sheen of dragon-like scales", but it seems like a reasonable ruling to allow.

Stone Sorcerers - Should be able to use either (if the beast has skin)

Per the discussion above about draconic resilience, if you turn into a beast that has skin you should be able to chose between the Stone’s Durability or the beast's AC.

Note: Stone Sorcerer is UA material so it is non-final.

The Elementalist Fighter class is more powerful at every level than a standard Battlemaster Fighter.

I'm using the Battlemaster Archetype as a reference point as the OP has stated that it is an influence on the homebrew class design and because the class features closely mirror that of the Battlemaster.

TLDR; Elementalist Fighter blends almost all of the features of a Battlemaster, adds a sprinkle of Barbarian and then adds a tonne of powerful abilities to boot.

• Second Wind comes 2 levels later, decreasing survivability at 1st and 2nd level, before returning to the status quo at 3rd.
• Weapon Bond is almost identical to the Eldritch Knight feature and comes 2 levels early. It somewhat undermines one of the EK's unique features but is not too major.
• Imbue Weapon is a reskinned Barbarian Rage, the quintessential Barbarian feature. It is more offensive focused than defensive. There doesn't appear to be an end condition, but I'll assume it's the same as Rage. Almost everything here feels out of flavour for an elemental focused fighter. Psychic resistance, temporary darkvision, changing the advantage on checks and saves to Charisma, and advantage on saves against being charmed are not, in my opinion, things I would associate with a class called "Elementalist Fighter". Mechanically, choosing the additional damage's type is strong as it somewhat enables low-level melee fighters to circumvent physical damage resistances, one of their greatest banes. It is somewhat comparable to the 3rd level Elemental Weapon spell, minus the to hit bonus and magical weapon consideration. Losing the physical damage resistance of Rage is not a suitable drawback, as Fighters are much more capable of achieving high ACs at low levels than Barbarians, through heavy armour, shields, and fighting styles. So the loss of defensive abilities is negated by the classes intrinsic high AC and the damage capabilities are greater than the standard Rage, with some odd extras thrown in.
• Maneuvers. Having extra restrictions on maneuvers is sure to only cause confusion. The save DC being Charisma based is also quite unexpected and again outside of the classes flavour. Some of the new maneuvers are far above the current power level. Elemental Slam stands out as particularly egregious. Fighters lack AoE as a balance requirement. Giving them access to AoE which does good damage for a relatively low resource cost, is definitely going to cause them to be overpowered relative to another fighter of the same level. In conjunction with the powerful secondary abilities that many of the damage type Slams offer, I don't know why you wouldn't pick this maneuver over any other.
• Ancient Blade doesn't have any real balance concerns. It does, however, cause a great deal more work for your DM, which is something that I would consider poor design. Know your Enemy of the Battlemaster is pretty whatever, but it doesn't cause any additional preparation or forethought that could be used on the game as a whole, instead of one character.
• Elemental Incarnation is absolutely insane. Indomitable is already a strong class feature, and EI blows it out of the water. It evaluates to casting one of the Investitures of X spells, without the need for concentration, once per day, at the cost of one level of exhaustion. And this is in a non-casting class, 2 levels earlier than a full caster could. The sheer damage output and/or utility that this feature could provide in its 10 minutes is incredible. Especially considering most conditions that require a save also give one each turn. Potentially missing a turn or two is well worth this.
• Elemental Explosion is again powerful because it gives Fighters access to powerful AoE at a relatively low cost.

If given the choice to play a Battlemaster or an Elementalist Fighter, the Elementalist is the clear winner. With its access to cheap AoE and additional damage output, it does everything a Battlemaster does and more. It suffers from a lack of identity, borrowing iconic features from other classes, and also having a mix of features that don't resound with the classes core concept.