Are the following sets a subspace of $\mathbb{R}^3$?

For the first you are on the right track. The sum of two vectors of the subset is in the subset precisely because the sum satisfies the equation $a+2b+c=0$, which you have proved fir $(a+a',b+b',c+c')$. Likewise for the product by a scalar.

For the second one, the equation $a^2=b^2$ is satisfied by both $(a,a,c)$ and $(a,-a,c)$. But if the subset $S=\left\{(a, b, c)^{T} \in \mathbb{R}^{3} | a^{2}=b^{2}\right\}$ is a subspace, the sume is also in it, that is $(2a,0,2c)$ for all $a,c$. But for $a\ne0$, the equation can't be satisfied. For instance $(1,1,0)\in S$ and $(1,-1,0)\in S$ but $(2,0,0)\notin S$, so $S$ can't be a subspace.