Question: $G$ is a PRG where $|G(s)| > 2 \cdot |s|$.

1. $G'(s) = G(s0^{|s|})$.
2. $G'(s) = G(s\_1...s\_{n/2})$, where $s = s\_1...s\_n$.

My question here is whether my own proposed solutions are correct and do make sense.

Answer: 
To expand on my comments, you need to define how $G$ is extended for inputs of different sizes before you can determine whether the resulting $G'$ are PRG or not.

1. With the definition in your answer that $G(s\_1 || s\_2) = G(s\_1)||G(s\_2)$ it is indeed not a PRG, but you *could* define e.g. $G(s\_1 || s\_2) = G(s\_1 \oplus s\_2)$, in which case $G'$ *is* PRG.
2. Again, it depends on the definition. Suppose $G(t\_1 || t\_2) = H(t\_1)||H\_(t\_2)$ for some PRG $H$ with a smaller input size. Now $G$ is PRG for random inputs $s = t\_1 || t\_2$. Define $G(t\_1) = H(t\_1)$ and $G'$ is PRG. However, if you define $G(t\_1) = G(t\_1 || 0^{|t\_1|})$ you are back to earlier point and $G'$ is *not* PRG.
In the first case, I would say that $G'$ is not necessarily a PRG because, following from the definition of a PRG, the input seed $w = s0^{|s|}$ to $G$ is not random, so the output can't possibly be pseudorandom: the leftmost $n = |s|$ bits of $w$ correspond to a random seed $s$, but the rightmost $n$ bits are all fixed to $0$. Furthermore, the $G(x)||G(y)$ concatenation $- \enspace x,y \leftarrow \{{0,1}\}^n$, $x \neq y \enspace -$ corresponds to the concatenation of 2 random looking strings; however, the $G(x')||G(y')$ concatenation, where $x' = s$ and $y' = 0^n$, doesn't correspond to the concatenation of 2 random looking strings, as the input seed $y'$ is not random.

In the second case, $G'$ could be a PRG because, following from the definition of a PRG: (1) [**expansion**] $|G(s\_1...s\_{n/2})| > 2 \cdot n/2 = n$, so $|G'(s)| > |s|$; (2) [**pseudo-randomness**] if $s$ is a randomly chosen seed, and $G$ is a PRG, then $s\_1...s\_{n/2}$ is a random string and $G(s\_1...s\_{n/2})$ is indistinguishable from a truly random string $y \leftarrow \{{0,1}\}^{l(n)}$, where $l(n) = |G'(s)|$.

In case there are any mistakes or not clear enough, please specify where am I wrong and argue why.