Ok, the first thing you have to specify is what does it mean for to talk about a Cartier divisor. There are a number of options here. The one I like best would be invertible subsheaves of $K(X)$, the fraction field of $X$.
Now, what do you mean by Weil divisors? One option is formal sums of points. This is ok, but there are problems in higher dimensions (mentioned below). Another option is $O_X$-module subsheaves of $K(X)$ in general (if you weren't dealing with curves maybe we should require these to be S2). Or maybe instead reflexive subsheaves? In higher dimension sometimes people require them to be Cartier in codimension 1. But this just gives us Cartier divisors...
Why is the first notion problematic? I'm not sure if how bad it is for curves over $\mathbb{C}$, but in higher dimension, the map from Cartier divisors to Weil divisors is not injective for non-normal varieties. If the same can happen for curves, there isn't a well defined notion of $O_X(nP)$. For example, if there are two Cartier divisors, which one do you choose? This is maybe ok for curves though, I'm not sure. I'll let you know if I see an example.
Now, let me answer your questions.
Yes, depending on what you mean by Cartier divisor. Embed your curve in projective space. Take a hyperplane $H$ (not containing $X$) passing through $P$ and some other smooth points on $X$. $O_{\mathbb{P}^n}(H) \cdot O_X$ is certainly invertible. Let $D$ denote the effective Cartier divisor on $X$ such that $O_X(D)$ agrees with $O_{\mathbb{P}^n}(H) \cdot O_X$ away from $P$. Now simply consider $O_{\mathbb{P}^n}(H) \cdot O_X(-D)$. This agrees with $O_X$ away from $P$, so I would say it is a Cartier divisor supported at $P$.
I don't think so. This doesn't work in higher dimensions either. I assume you want some properties? Like preserving linear equivalence?
If $O_X(nP)$ is not well defined, then I don't think you can define the notion of $\mathbb{Q}$-Cartier. I guess you could say that there is some Cartier divisor that maps to $nP$ in the natural map from Cartier divisors to Weil divisors?