Is $L:\mathbb P_4\to \mathbb P_5$, with $L(p)(x)=\int_0^x p(s)ds$, a linear function? If we consider a similar question, $L(p)(x)=7+\int_0^x p(s)ds$,

We can definitively say no. 

Let $p(x)=q(x)=0$ be the zero polynomial. So we also have $p(x)+q(x)=p(x)$. We compute $$L(p)(x)=7+\int_0^x 0 ds = 7$$ But since $7+7\neq 7$ we have $L(p)+L(q)\neq L(p+q)$ so the additivity rule is not satisfied.

But if we remove the 7, does it still remain not a linear function, or does it become a linear function?
Let $p,q$ be in $\mathbb{P}_4[\mathbb{R}]$ and $\alpha, \beta \in \mathbb{R}$, then:

$$L(\alpha p + \beta q) (x)= \int_0^x \alpha p + \beta q ds = \alpha \int_0^x p(s)ds + \beta \int_0^x q(s) ds = \alpha L(p) (x) + \beta L(q)(x) $$
So $L$ is linear.
Let $p,q$ be in $\mathbb{P}_4[\mathbb{R}]$ and $\alpha, \beta \in \mathbb{R}$, then:

$$L(\alpha p + \beta q) (x)= \int_0^x \alpha p + \beta q ds = \alpha \int_0^x p(s)ds + \beta \int_0^x q(s) ds = \alpha L(p) (x) + \beta L(q)(x) $$
So $L$ is linear.