HINT for (2): Assume that $X$ is infinite. First show that there is a family $\{U_n:n\in\Bbb N\}$ of pairwise disjoint, non-empty open sets. One way to do this is to pick a point $p\in X$ that is not an isolated point, and let $V_0$ be any open nbhd of $p$. Pick $x_0\in V_0\setminus\{p\}$, and let $V_1$ be an open nbhd of $p$ such that $\operatorname{cl}_X V_1\subseteq V_0\setminus\{x_0\}$. Continue in the same fashion: given $V_n$, pick $x_n\in V_n\setminus\{p\}$, and let $V_{n+1}$ be an open nbhd of $p$ such that $\operatorname{cl}_X V_{n+1}\subseteq V_n\setminus\{x_n\}$. For $n\in\Bbb N$ let $U_n=V_n\setminus\operatorname{cl}_X V_{n+1}$; I’ll leave it to you to verify that the sets $U_n$ are pairwise disjoint and non-empty.
Now for each $n\in\Bbb N$ let $f_n:X\to[0,1]$ be a continuous function such that $f_n(x_n)=1$ and $f_n(x)=0$ for all $x\in X\setminus U_n$. (How do I know that there is such a function?) Now show that the functions $f_n$ are linearly independent.