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.
This shield is very powerful
By the first benefit alone, the shield is comparable a Shield +2. Having half-cover grants +2 AC and DEX saving throws. The reason this is not strictly better than a +2 Shield is because it does not stack with other sources of cover (and can be nullified by features that ignore cover). A +2 Shield is a rare magic item which according to DMG guidelines is appropriate for characters of level 5 or higher.
The second benefit as you note is extremely good. Spending half of your movement is a negligible cost in a lot of situations, especially on a paladin. With access to Find Steed or other mounts, you aren't using your own movement for anything else so spending half your movement to get 3/4 cover is almost free. Even if you don't have a mount, you may not need to move over half your movement often enough that you get the chance to activate this. Having +5 AC for a round is comparable to the Shield spell. Yes, this is just a lvl 1 spell but it requires a reaction to use and usually expends some sort of resource (a spell slot or charges on a magic item)
The third option alone is actually not that strong. Expending your action to grant an ally half-cover is a fairly weak option when standing in front of your ally usually already does this. Having an extra option for an action doesn't really hurt though and in a situation where you can't reach your opponents and have nothing else to do it might be useful.
All of these features together make the shield an incredibly powerful option, especially considering that you can purchase these for 75 gp. I suspect anyone with 15 STR that can use a shield would probably use one.
Is this actually a problem?
Whether having this item in the game is actually an issue is doing to be very group dependent.
For example, if all the PCs have access to very powerful items it might just be fine. The DM may just be planning to throw more powerful encounters at your party or just wants to run a game where the PCs can win encounters fairly easily.
Having a very high AC also does not mean you are invincible, see this related question.
The DM may simply not have realized how powerful this shield is
This can happen often. The DM hands out an item without realizing just how powerful it is until it actually comes into play later.
The important thing to do is to assume good faith on the DM's part. Don't accuse the DM of handing out an OP item. Instead, try to explain how this might affect the game in ways that may not be fun for the group. Let me give you an example:
In a homebrew campaign I played in, after defeating some orcs of Gruumsh, the DM allowed us to keep their blood and apply it to any weapon for a permanent +1d8 damage (like the orcs' trait). While it seemed fine at the time, an extra 4.5 damage per hit for the martial classes was enough to feel unfair on the wizard and bard, who were now resorting to using weapon attacks instead of cantrips. So, after discussing this with the DM it was reduced to 2 extra damage and limited to mundane weapons. I honestly still felt it was a bit much at the time (I tend to be a bit of a balance maniac) but I let it go and it turned to be fine during gameplay.
My takeaway from this is to let it play out a session or two and see what happens. If it is causing actual issues and causing the game to not be fun it should be fairly noticeable and you can then approach the group to discuss how to go about improving the situation.
Maybe it is agreed that the item should be nerfed. Maybe the DM just hands out similarly powerful magic items to the other PCs and cranks up the difficulty of the opponents. Or maybe something else entirely.
Best Answer
Is it overpowered? As you've written it, perhaps.. but only because you worded one of your abilities to always knock something prone, this is problematic.
If a character passes a save you can't and shouldn't knock them prone. This is bad design as a save should let the character in question avoid the worst parts of an effect. In this instance, the lesser of two evils (this being the knockback effect) would likely be applied, perhaps in a smaller denomination compared to a failure. Perhaps something along the lines of "Prone and 20ft knockback on a failure" and "10ft knockback on a success". This is fair enough for your PCs to come up with some creative uses for it (ship combat comes to mind) and if its turned against them they'll likely still feel that it's fair enough (aka I'm further away but at least I'm not prone.)
Generally, to determine if something is overpowered, we have to gauge it against something else already that already exists within the scope of the game and evaluate the merits based on something similar.
So currently you have:
Currently the closest spells to emulate these effects are:
These are all things a ninth level wizard would be able to do without any trouble with better effectiveness. If you removed the last effect, A third level wizard would be able to duplicate the effectiveness of this weapon.
If you want it to be more balanced. Amp it up some but remove the auto-prone. Give the weapon an enchantment bonus, and ten charges that refresh daily after a long-rest. Make the first level effects cost one charge, the second level effect cost two charges, and the fifth level effect cost five charges to limit how much you can do with it per day.
Also, Since the DC12 is a pretty low and very easy save to beat at tenth level for most things, you could make it so that for every three charges you use you could increase the save DC of the weapon's prone / knockback effect by one, which would bring the save up to around DC15 if you were in a pinch and needed to burn the charges to get something away from someone / knock someone down.
As a balancing mechanism you could even go so far as to have the weapon lose its enchantment bonus for the day if all of its charges are consumed.
So as far as the weapon goes, here's a general Stat block:
Kusanagi no Tsurugi
Weapon(longsword +1), Very Rare 10/10 charges
This yellow bladed saber with a green hilt can control the wind, sending a gush of strong wind in the direction of the strike of the sword. Kusanagi has a fine edge, tempered in the wind, and as long as the weapon is imbued with magical strength, it retains its +1 Enhancement bonus. When all of Kusanagi's charges are used, the weapon becomes a non-magical longsword until the user takes a long-rest in a place where wind is present.
As an action, when drawing this weapon from the sheath, a slash of this sword sends a 60 foot cone of wind in any desired direction. Any creature inside this cone must make a DC 12 strength saving throw being pushed back 20 feet and falling prone on a failed save or being pushed back 10 feet on a successful save. This pushes fire out of the cone's area unless it hits a wall, in which case it snuffs out. It also disperses fog. Air elementals and other non-corporeal creatures have disadvantage on this saving throw. (1 charge, +1DC/3 charges)
Kusanagi no Tsurugi can also create the following magical effects, provided the weapon has charges:
Kusanagi no Tsurugi is a blade made of and forged by the wind, in places where no wind is present such as deep underground or underwater, the charges of the weapon cannot be used.