What exactly interests you? Let me discuss $n=20.$
The number of primes below $2^{21}$ is evidently $155611$ of which $73586$ have exactly 20 bits. Given that, the answer to your question 2 with $n=20$ is definitely yes for $m$ below $73586$ and no for $m$ above $155611$. Doing question 1 for $m=70000$ would be tedious. If you wanted $n=20$ and $m=481$ then one could find a $k$ , a $20$ bit multiple of $2*3*5*7*11$ and take it along with $k+j$ for the $480$ numbers $j$ less than $k$ and relatively prime to it.
It might be more interesting to ask about finding $m$ numbers between $s$ and $t$ when $t-s$ is "small" with respect to $s.$ The case $s=2^n$ and $t=2^{n+1}$ seems too broad.