Unfortunately I don't have enough reputation to vote up your answer.

Aha! Somehow your previous Jacobsthal reference missed my attention. Not that I understand it immediately, but I have confirmed that your $k$ values $1,1,2,6,8$ give better coverage indeed. This confirms the answer to question 2. is "no" and 3 is "yes". Thanks a lot.

Restating @Gerry Myerson above: Given any $r-1$, let $p$ be a prime dividing $r$, then $r-1$ is in the arithmetic progression $k+np$ with $k=p-1$ which covers up to $N=p_{m}-2$ in the case described above. Now extending on that: Given any $r-t$, let $p$ be a prime dividing $r$, then $r-t$ is in the arithmetic progression $k+np$ with $k= (p-t) \bmod p$ which also covers up to $N=p_{m}-2$ if $t$ is odd. Unfortunately this includes some $k=0<1$ so $t=1$ is the only acceptable value of $t$.

Added the third question because if the answer to 2 is "no" because if the coverage is more than up to $N=p_{m}-2$ it is interesting to know how much more.

Yes $m$ distinct consecutive primes and each is a difference in exactly one of the $m$ arithmetic progressions.

@The Masked Avenger thank you for the suggestions. After taking a quick second look at Tao, Green, Ford and Konyagin I found out that maybe it is near this than I thought, but surely I have to read it few times more. One question though does your answer "Q2: no." hold after I added the restriction that the arithmetic progressions are exactly one with each prime?

@Gerry Myerson as I said it was a good "quick start insight" giving me something to think about. Having done that I understand as all numbers up to $p_{m+1}$ are divisible by one or more prime up to $p_{m}$ your suggestion shows that the numbers are covered with this kind of arithmetic progressions. Thanks again.

@Włodzimierz Holsztyński I have refined the text again, even though I thought the first prime was included in previous revision I agree some parts were not as clear enough. Yes the primes are the first $m$ primes including 2, and one and only one arithmetic progression with each one. Thanks again.

I have updated the question in an attempt to make it more self-contained (and correct grammar). I hope I have understood your suggestions Wlodzimierz. Your input Gerry is very helpful, it gives a "quick start" insight. Thanks to you both.