Undecidable: $w$ on which a TM M $M$ halts after $\leq w$ steps The detailed question is:

Is there a word $w$ on which a TM M $M$ halts after a maximum of $|w|$ (word length) steps?

I highly assume, that this problem is not decidable because in the worst case you have to test every word that exists (infinite) to realize, that the TM never holds. However this isn't a proof and I have to proofe this question (without the Rice's theorem).

My idea was to use the subroutine technique to "convert" the problem into the halt-problem, the $\epsilon$-halt-problem or the total halt problem. I have no idea on how to turn this problem into an existing one - could you help me here please?
Let $M$ be a Turing machine and consider the Turing machine $M'$ that simulates $M$ on an empty input (regardless of the input to $M'$).

If $M$ halts on empty input then $M'$ eventually halts too. This means that there is a word $w$ of sufficiently large length $|w|$ such that $M'$ halts after at most $|w|$ steps (actually, there are infinitely many such words).

If $M$ does not halt, then also $M'$ doesn't halt (regardless of the its input). 

Therefore your problem cannot be decidable as otherwise there would be a Turing machine capable of solving the halting problem.