In white: e3. Checks on the king are met with the king capturing the checking piece, and other moves lead to white putting a pawn on d4 or f4. If black takes on f3, the e pawn is the one that moves on move 2.
Mate in two(37)
correct
Again, I can't find a flaw unless you consider the 4 checkmate threats a flaw. But to my knowledge, that is not a flaw but a theme (Fleck). Black can defend against every 3 of them but is then mated by the 4th.
Of course there is lots of other stuff as well. This probably counts as a record for K+P checkmates! Great problem!
I think I know the "flaw". There has to be a promoted white unit on the board. As a retro-specialist it does not bother me a lot but I suppose some judges deduct for that.
Interestingly, the white promotion can be fixed but the resulting position then features a black promotion. Not much of an improvement.
Edit: Well, not quite true. Provably a promotion has taken place but it is not sure that the promotion piece is still on the board! I can't really classify that as a flaw.
I certainly do not know,or would I really care about any aesthetic flaw,but he again just wrote,"The record of direct mate by multiple WP without promotion is six,triumphantly combined in 37† with WK6". I am thinking that you enjoy trying to find the flaw and tweaking them, and I like reading your responses,so I will post some more and put the same dagger mark he uses on the flawed ones.
Personally, I am not all that concerned about aesthetic flaws either. An aesthetic deduction on one item may be compensated and more by plusses on another. And the problem then still remains prize worthy though that doesn't concern me a lot either. An aesthetic flaw might be something like getting 7 out of 8 moves of a knight wheel, non-essential duals as in the K+Q record problem or using lots of cookstoppers (a cookstopper is a unit in the diagram with the exclusive function of preventing a cook or dual).
Much worse are the technical flaws, violations of the composition code such as "unprotected checks in the diagram", "a king checking key move", "taking a kings flight square in the key", "capturing a piece in the key". "dual checkmates in a thematic line" and such things. Note that Petrovic problem with the two knight wheels is flawed technically and would be severely penalized for that in a competition. Without the flaw, it would be 1st prize in any contest.
It is unclear to me if the author of your book only marks what I call technical flaws or also aesthetical issues. Also I don't know which criteria he uses for both categories. They may be different from mine. If it is an old book, it might also reflect the standards in a different era. So I am somewhat in the dark as to what to look for behind his dagger marks!
Note that the argument behind the composition code is to prevent cheap and ordinary compositions. It therefore frustrates you in playing the percentage moves of standard chess like; "capturing material", "checking the king", "encircling the king" (taking flight squares). Composition chess is to a high degree the inversion of game chess. Read the discussion between Rocky64 and Polar-Bear in the "an introduction to composed chess problems" post to see that one of these two doesn't understand that 
White in two by K. H. Hannemann 1932