PRF and deterministic polytime function implie PRG
For your points:

Your discussion of expansion is correct. In particular, you can show that $\ell(r) = mr \geq 2r &gt; r$, which is essentially what you explain.

You can write a formal proof that $F_C$ is poly-time in essentially the same way you would show any algorithm is poly-time. In this case, it is quite obvious that the result is poly-time, so if it feels obvious to you that this is the case that should be normal. That being said, there are some subtleties --- concatenating two things with polynomially-sized representations is poly-time, but what if you have exponentially many things? This would no longer be poly-time. Can you be sure the algorithm is still poly-time in light of this (and if so, why?)

You have a definition of pseudorandomness for PRGs (definition 3.14), which likely want to relate to a definition of pseudorandomness for partial function (in particular $F$). I would suggest finding the particular definition you're working with first. Your proof strategy should be reduction to $F$'s pseudorandomness -- some statement along the lines of &quot;Any distinguisher for $F_C$ implies a distinguisher for $F$, which as $F$ is pseudorandom does not exist, so the distinguisher for $F_C$ cannot exist as well&quot;. This is the &quot;general template&quot; of proofs by reduction within cryptography, so you would do well to internalize this &quot;strategy&quot;.