Let $X\subset \mathbb{P}^n$ be a complex smooth projective variety of degree $d$ and dimension $m$, such that $X$ is not contained in any projective subspace of $\mathbb{P}^n$. Let $P\subset \mathbb{P}^n$ be a projective space of dimension $n-m$. Then we know that if $P$ and $X$ intersect properly, $P\cap X$ is a zero-dimensional subscheme of length $d$.

I'm wondering if some similar properties hold for non-properly intersections. More precisely, I have the following questions.

Question: Suppose that $X$ and $P$ do not contain each other, and $\dim P\cap X= 1$. Assume that $n$ and $m$ are not too small (for example, take $n=9, m=6$).

Is the projective scheme $P\cap X$ pure-dimensional? I guess the answer is yes, but I don't know how to prove this yet...

How to compute the degree of $P\cap X$?

When is $P\cap X$ connected?

I think one of the possible methods is to describe $P\cap X$ as the zero locus of a section of $\mathcal{O}_X(1)^{\oplus m}$.