my experience today

its would be nice puzzle 

It's a good game combination but it wouldn't make a good puzzle as white has a choice between two moves mating quickly.

Fine for a puzzle, mate in 2 is clearly better than mate in 4.

There is no mate in 2 and not even in 4!

I think both first moves are mate in 5 but I am not completely sure of my calculations.

You're right, I overlooked a futile interposition in one case (marked with asterisk below) and a rather good interposition in another.  Mate in 3 in the first case, and my second idea may not work at all.

1. Rg8+ Rxg8

2. Qxf6+ Rg7* 

3. Qxg7#

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My second line started:

1. Rxh7+ Kxh7

2. Rh4+ Kg7

4. Qg4 and here I missed ...Qg6 where the attack fails

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But there's always :

1. Rxh7+ Kxh7

2. Qf5+ mates quickly, e.g.

  2.   ...Kh8  3.  Rh4 Qh5 4. Rxh5 Kg7 5. Qh7#  but black can stretch it out a bit.

One more idea:

1. Qf5! threatening Rxh7# or Qxh7#

 1.  ...Qg6

2. R7xg6 and mate follows, in 3.

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A similar idea is Qb7! , I believe that's a mate in 4. (two futile interpositions on f7)

Ah, there is more to it than I had seen! My main lines were 1. Rxh7 and 1. Qf5. I completely missed 1. Rg8+ which is definitely stupid.

Well done!

(wonder though what the OP had in mind)

Very simple Rg8+, Rxg8, Qxf6+, Rg7, Qxg7# game over

Quite simple