As you asked for hints I will not go into full detail. I assume that $C([0,1])$ refers to the real vector space of the realvalued continuous functions.
Notice that the map $\phi \colon X \to \mathbb{R}$ with $\phi(f) = \int_0^1 f(t) \,\text{d}t$ is linear and continuous, and that for
$$
S = \{f \in X \mid \phi(f) = 0\}
\quad\text{and}\quad
T = \{f \in X \mid \phi(f) \neq 0\}
$$
we have $S = \phi^{-1}(0) = \ker \phi$ and $T = \phi^{-1}(\mathbb{R} \setminus \{0\})$. From this you can answer 1a, 1b and 2a immediately.
For compactness of $S$ pick some $x \in [0,1]$ and notice that the evaluation $e \colon X \to \mathbb{R}$ with $e(f) = f(x)$ is continuous and linear. What do you now about the image of a compact set?
For connectedness of $T$ similarly consider $\phi$ and think about the image of a connected set. (Here we use that we are only considering realvalued functions.)
For the denseness of $T$ notice that $X = T \cup S$. Can you (uniformly) approximate elements of $S$ by elements of $T$? (Hint: How does $\phi$ react to moving functions up and down?)
PS: Some more deatails regarding the compactness of $S$ and connectedness of $T$:
Take some non-zero $f \in S$. Then there exists $x \in [0,1]$ with $f(x) \neq 0$. So for the corresponding evaluation map $e_x \colon X \to \mathbb{R}$, $h \mapsto h(x)$ we have $e_x(f) \neq 0$. Thus $e_x(S)$ is some non-zero linear subspace of $\mathbb{R}$ (which there are not many of). If $S$ was compact then $e_x(S)$ would also be compact, because $e_x$ is continuous.
For the denseness of $T$ notice that for every $f \in S$ and $\varepsilon > 0$ we have
$$
\phi(f+\varepsilon) = \phi(f) + \phi(\varepsilon) = \varepsilon \neq 0
$$
(here we identify each constant $c \in \mathbb{R}$ with the corresponding constant function on $[0,1]$), where we have $\|f - (f+\varepsilon)\|_\sup = \varepsilon$.
