(Puzzle #2):
Loser Academy
Effectively this type is very similar to the Helpmate type in standard chess compositions. In puzzle 1,2 and 4 they are the same with reversed colors. In puzzle 1 all queens but "Qb8" are redundant since black must play the best move anyway which is to checkmate by capturing the queen. No need to encircle the black king. Puzzle 4 is flawed as white can also start with 1.hxg8=B/N. Puzzle 3 is also flawed since white can start with 1.Kb1! as 1. ..Qh7+ is the best response; after 2. Kc1 Qc2# we have the same result.
Btw, this would repair puzzle #4:
The solution for #6 looks OK but the position is illegal due to all the promoted white pieces plus the white pawns.
I did know the position is illegal in the strict sense of the term in a normal chess game, went over-the-top with the queens, but I care more about doing a good and fun puzzle with a correct solution than if the initial position can be encountered in a normal game due to material.
I didn't think puzzles absolutely needed to be something you can take from material from a normal game or else they are invalid.
This is the same philosphy I had with The Bishoper's wrath, the White bishop could not end up where it was in a normal game.
But, in terms of puzzle compositions, I am not ok with (in two-players game):
a) bad continuity: if it is player B's turn, player A must have been able to play a legal move before
b) pawns in illegal positions (1st or 8th rank), multiple kings or missing kings
c) player A's king in is check on B's turn.
d) a player is in triple-check or more
e) touching kings
f) Player A is in double-check in a way it can't have resulted from player B's last move.
The solution for #6 looks OK but the position is illegal due to all the promoted white pieces plus the white pawns.
Is «Chess.com analysis engine doesn't disapprove the position as being invalid» a good enough criteria for making a puzzle, or is it not? Or is there a consensus in the puzzle community about it I must know?
I do know the engine analysis accepts to load position like these (White to play):

The solution for #6 looks OK but the position is illegal due to all the promoted white pieces plus the white pawns.
Is «Chess.com analysis engine doesn't disapprove the position as being invalid» a good enough criteria for making a puzzle, or is it not? Or is there a consensus in the puzzle community about it I must know?
The consensus in the chess composition community is that in problems where the standard rules apply, the position must be legal. Here "legal" means it's possible to reach the position from the opening game array (it's irrelevant whether such moves are plausible).
The reasons for illegal positions range from the very simple, like two adjacent kings, to the extremely complex. The more complex arguments for why a position is legal/illegal involve a process called retro-analysis, or logically deducing what must have occurred in a hypothetical game.
When setting up a position for an engine, it checks whether it's "valid"; "invalid" positions are the simplest kinds of illegal positions, e.g. adjacent kings, missing king, pawns on the 1st/8th ranks. Detecting such invalid types is easy to program with a few rules, but an engine is utterly incapable of doing retro-analysis to determine the non-trivial kinds of illegality. Hence you're allowed to set up positions that even basic retro-analysis would've shown as illegal, e.g. any position with WPb2 and WBa1. Hence the "invalid" check provided by an engine is an insufficient tool to decide what's acceptable in problem compositions.
Ah, I missed the one in the "Bishopers wrath" But indeed, it is illegal. Rocky64 explained the rules nicely. Here is some more:
In problems like the "Bishopers wrath" you could add the "Chess960" condition which places the puzzle slightly outside standard chess. Not that much though as this type increasingly shares the market with standard chess and almost all of its rules are the same. At this time however there are no (not yet) conventions about which castling rights to permit for Chess960 positions. which may lead to ambiguities.
What is interesting is that the exclusion of illegal positions is in the FIDE game rules (article 3.10.3) and does not originate from the puzzle/composition conventions. People may claim that "puzzles" are not the same as "compositions" as only the latter are formally regulated. But when I ask what the "puzzle rules" are about e.g. "e.p." and "castling" rights they go silent. Such rules are necessary to complete the play options in chess diagrams. In the absence of an organization for "puzzle rules", I (and most of us) choose to treat all puzzles as compositions with regard to missing information instead of choosing for chaos.
The legality requirement for promotions is an issue for compositions with abnormal material distributions (like 8 queens) and many pawns but not for your creations. There is lots of randomly placed material in them which appears to have only an aesthetical function and may be eliminated or replaced by more efficient artillery pieces. Composers take the legality requirement as a challenge to demonstrate their skills in that department. How many great compositions do you think were discarded due to the absence of the "a0" square on a legal chess board?
At Loser Academy, we learn that there is no point in winning
! As White, find the worst moves possible to ensure you lose as fast as possible (i.e. mate in one is better than mate in two) Black plays normal and logical moves, the best ones in the position. Feel free to share your puzzles!
(Puzzle #1):