[Note that saying 'check' is something amateurs do; in official games this is even forbidden. In that light it seems the following convention could be useful for this game at amateur level, to help enforcing the checking rule (which might strike ordinary chess players as unusual):

A player that attacks a castle and suspects it might be checkmate can say 'check'. This obliges his opponent to point out at the end of his next move the Castle where his King takes shelter. If that Castle cannot be captured, the game just continues, because even if there was a genuine check it has apparently been resolved. If it can be captured, the player that pointed it out could be ruled to lose, or (in a more forgiving spirit) could be obliged to acknowledge his mistake and point out another Castle (or conceed he has been checkmated or played an illegal move).]

• I agree we don't need to say check. Just need pay attention to checkmate, because of course if you are in checkmate, the game is over and all other move is an illegal move.
• I played a few matches with the obligation to point out every Castle under attack and I did not dislike it, but sometimes we moved a piece and did not realize that the move resulted in check on some of opponent's Castle, and I did not like can't be able to attacking this Castle for not having said check, it totally distorts game.
• Turnover allows you to checkmate in more than one Castle at same time. I've already finished a match with 3 Castles in checkmate and it's nice to talk check and check again, seeing the opponent's face when realizing there are a lot of Castles in check. I would have to test this possibility further. 
  • See it on: https://www.instagram.com/p/By00QZ9DhoG/
     

[As to notation: this is not really part of the game rules, and for orthodox Chess several systems exist (descriptive notation, coordinate notation, long/short algebraic notation). I don't particularly like the notation proposed here; personally I would just stick to an existing standard, like SAN. (Also for the reason of being able to process it with existing software.) This is based on identifying a move by mentioning the type of the moving piece and the target square, possibly augmented by as many coordinates of the starting square as is needed to distinguish it from other legal moves that would be witten the same without disambiguation.]

• I agree. Notation is a world and I just bring to myself a version to be able to take notes of my test matches to be able to animate it after. So I have learned a lot about notation and created mine. Of course each player will use his own notation system, its not a rule.

[Turnover Chess offers two new aspects one usually does not have to deal with: piece types are 'fluid', and besides capture there are the possibilities of (friendly) combination and turnover. in SAN one makes the distinction between capture and non-capture by inserting an 'x' symbol, (problematic on board with more than 23 files, but then, more than 26 files would be problematic anyway...), although this is redundant, as it is fully implied by the occupation of the target square. Apparently it is considered so helpful to make this distinction jump out that it is worth the redundancy. This would then likely be true for combination and turnover as well.]

• Well, the two points ":" I used to highlight a turnover, was just marketing :) Thinking in a future where two points is a kind of trademark, icon or turnover brand. Its not essentially needed. We don't need to differentiate if it is a friendly combining or a turnover.
• x mark is so cute :( it gives dopamine :) when you take an opponent piece. I agree it can be redundant. If I take, instead use x to indicate a take, just notate which piece is seated on square gives the same result and x will be unnecessary. I will try notate some matches to see it better.

[My first idea was to write any move to a non-empty square with 'x', but that would not distinguish combination / turnover from true capture. It is possible to make this distinction within the existing framework of SAN, however, by using promotion suffixes: any move that would not lead to the expected occupation (i.e. the outermost ring of the moving piece) of the target square could be written as a promotion to what does result. The use of 'x' would be determined by whether the former occupant of the target square was friend or foe. So Qe3 would be a non-capture, putting a Bishop on e3, while Qxe3 would in addition have captured an enemy there (Knight, Queen, Bishop or Castle). Qe3=Q would have absorbed a frienly Rook, Qxe3=K would have turned over an enemy Pawn. In all cases a Rook would be left behind, but that is implied in this game, and so common that it does not warrant introduction of more redundant symbols in the notation.]

• Just write the result is good. Instead QxQe3, just Q.e3 as I write or Qe3 is fine, but less exciting in therms of x as "hey have you seen? I take it!". I use a point to diferenciate move and combination, Q.e3 = "Queen moved to e3" so it is its Bishop going to e3 and Rook left behind... and Qe3 = "A Queen was mounted in e3", so evidently there is a Rook in e3 to be able to perform Qe3. I like it.

[Some more remarks: I think that using K for Knight is a really bad idea. It is mega-confusing to chess players. Even if you don't have to use 'K' for indicating another piece type, it would be better to keep N for Knight. After all, you kept the names of the orthodox pieces, while they are really not pieces at all but just constellations of rings. You might as well keep their standard abbreviations.]

• I like it. N is better. Friends said I need keep distance from chess, so I suppose use K was a way to try keep this distance, but N is good. 

[Not distinguishing Castles and Pawns in notation is probably also undesirable (even if they move the same). So I would always mention a Castle as C, (a good reminder for what is left behind), while Pawn moves could follow the normal SAN rule of using no type indicator at all, but always prefixing a 'x' symbol with their file coordinate. This remains unambiguous here, as Pawns can only have come from 3 squares to reach the mentioned target square, and only true captures can change file.]

• I have tested it a lot, and it is really not needed to diferenciate C and pawn. When I start a match I just write a3 and it is a Ca1 goint to c3.