Yes, a generic curve with one node at a fixed point of $\mathbb P^2$ over a field of characteristic $0$ has only that singularity. This is true in any degree $d$ at least $2$. By looking at a generic union of $d$ lines, two of which pass through a point $p$, we see that a generic element of the linear system of curves or degree $d$ with a singularity at $p$ has a node at $p$, and no other fixed points. The result follows from Bertini's Theorem for linear systems.

Yes, a generic curve with one node at a fixed point of $\mathbb P^2$ over a field of characteristic $0$ has only that singularity. This is true in any degree $d$ at least $2$. By looking at a generic union of $d$ lines, two of which pass through a point $p$, we see that a generic element of the linear system of curves or degree $d$ with a singularity at $p$ has a node at $p$, and no other fixed points. The result follows from Bertini's Theorem for linear systems.

[Edit:] for a more elementary approach, it is easy to show that the curve with equation $xyz^{d-2} + x^d + y^d$ has no singularities outside of the obvious one.