I assume it wouldn't feel pointless to you if it had been presented as a mate in 4 problem. (A hint to the composer.)
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I shall appreciate a response./comments. Again, thank you.
The problem is faulty because White has three first moves to force mate in 6 or less: g6, Bb6, and Bd8. You gave the winning line for 1.g6, although your final move should be Ngf3#. You didn't address 1.Bb6. You gave one defensive move against 1.Bd8, so let's look at it first.
1.Bd8 g6 2.Bf6 h4 3.Nh3. Now the black knight has to move, and if it doesn't capture the f6 bishop, White plays 4.Ne5+ Kd4 5.Ng4+ Kc4 6.Nxe3#. That leaves 3...Nxf6 4.Nf4 h3 5.gxf6 g5 (or h2) 6.Bd5#.

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Nice idea, but do you realize 1.Qg5 works just as well?
Yes, but it's not the fastest mate.
Yes, that may be correct, but a position that is easily won by quite a caveman move available feels a bit pointless as a puzzle to me - but that may be just my personal taste.