Problem 1: $\S$ $=$ $\{$ $(-\infty,a)$ $:$ $a\in \mathbb{R}$ $\}$ $\cup$ $\{$ $(a,\infty)$ $:$ $a\in \mathbb{R}$ $\}$ is a subbasis for $\mathbb{R}$ with the standard topology.
Attempt: Clearly, $\S$ covers $\mathbb{R}$. I must show that $\tau$ (the topology generated by a subbasis) is the standard metric topology. Indeed, since every open interval can be expressed as the intersection of members in $\S$, and each open set with respect to the standard metric topology is the union of open sets, the result follows.
Problem 2: Given $\S$ in the previous example. Show that $\S_n=\{\pi_i^{-1}(S): S\in \S, i=1,2,..n $ $\}$ is a subbasis for $\mathbb{R}^n$.
My attempt: To see this, wlog assume $n=2$. I must show that $\mathbb{R}^2$ =$\bigcup_{S\in \S_2}$ $\bigcup_{i=1}^2 \pi_i^{-1}(S)$ . Let $(x,y)\in \mathbb{R}^2$. Then, $x\in \mathbb{R}$ and $y\in \mathbb{R}$. By the previous example, $x\in S_1$ and $y\in S_2$ for some $S_1,S_2\in \S$. Hence, $(x,y)\in \pi_1^{-1}(S_1)\cup \pi_2^{-1}(S_2)$. Therefore, $(x,y)\in \bigcup_{S\in \S_2}\bigcup_{i=1}^2\pi_i^{-1}(S)$. Now I must show that the standard metric topology is the same as the topology generated by the subbasis. To do this, it suffices to show that $\mathbb{B} =$ $\{$ $\bigcap_{i=1}^nS_i$ $:$ $S_i\in \S_2$ $,$ $n\in \mathbb{N}$ $\}$ is a basis for the standard metric topology on $\mathbb{R}^2$. In which case, the result will follow, since the topology generated by a basis is unique. By continuity of projection maps, each member of $\S_2$ is open in $\mathbb{R}^2$. Since every member of $\mathbb{B}$ is the finite intersection of open maps, it follows that $\mathbb{B}$ is a subset of the standard topology. Since projections are open maps, if $U$ is open in $\mathbb{R}^2$ then $\pi_i(U)$ is open in $\mathbb{R}$. Hence $\pi_i(U)$ is the union of finite intersection of some sub collection of $\S$. Hence the result follows.
Are my attempts correct?
For problem 1 you only showed that all open sets of the standard topology are also open sets generated by §. To show that they are the same you also have to show that sets that are not open in the standard topology are also not open in the topology of §, or equivalently that all open sets generated by § are also open in the standard topology.
For problem 2, the things you detailed are all correct, as far as I can see, but the final “Hence the result follows” I cannot see. You definitely need to detail out why you think the result follows.