How helpful is high-diving if you want to become a better competitive swimmer? They are separate competitive Olympic sports, which both happen to take place in a swimming pool. In chess, both game-playing and problem-solving are recognized by FM, IM and GM tiers of certification.
See the attached chessbase report of this year's world chess solving championship. All the British team at least are also long-standing over-the-board GMs.
Answering the original questions directly:
My own experience is that designing & solving helpmates & retros has probably been detrimental to my over-the-board skills. It's all too easy to leave a piece hanging because the assumption at the back of my mind is that the opponent is co-operating!
Perhaps if I had focused on direct mates and endgame studies, there might have been more improvement in my game-playing. I agree with another reply about the likely value of direct mates in 2 in moderation for developing short-term analysis skills.
(2) Time allocation?
For a mate in 2, I aim for about 10 minutes, but it's quite variable. Definitely having a time in mind helps me focus. Longer problems take longer, of course, but it's not exponential: for mates in 5+, there is usually a single solution path to be found, which is iteratively defended against by the opponent.
There are definitely techniques for solving composed problems faster, but those often have little applicability to over-the-board chess. For example, if there is a long line to be evaluated, which clearly can begin with either of two alternative moves, that the line can be immediately dismissed, because if the problem is sound, there will be just one unique solution.
Another technique is to look for artistic effects which seem to be embedded in the piece arrangement. For example, is there a key square which several pieces (e.g. pawn, rook, bishop) might occupy in separate lines in order to block the other two pieces? This so-called Grimshaw theme could well be main point of the problem.
It would be total madness to analyze a game in this way, so these tricks are not convertible to over-the-board chess.
Final point: the same certifications (FM, IM, GM) are also available for chess problem composition. Not sure this has any direct analogy in the swimming world!