Homology and Homotopy in the Plane

Question

Suppose we're living in the plane minus (possibly infinitely many) isolated points, which I'll call poles. Intuitively, the following two statements seem reasonable:

1. Loops in the plane are homotopic iff their interior contains the same poles.
2. Loops in the plane are homotopic iff they are homologous.

Are these statements true? Why and/or why not?

Edit:

I forgot about winding numbers, so I'd like to expand on statement 1: Does it hold if the winding number of each pole is no greater than 1 in absolute value, and the loop does not intersect itself? Same goes for statement 2.

Edit 2:

I asked about the edited version of question 2 separately.

Answer 1

Regarding question 1, even two non-self-intersecting curves that enclose the same sets of points with winding number 1 might not be homotopic:

Answer 2

The first statement is not true, even for the plane minus finitely many (even one) point. Assuming we know that $\pi_1(\mathbb{R}^2\setminus 0) = \mathbb{Z}$, generated by loops around $0$, then the double winding around $0$ and the single winding around $0$ both contain the same "pole" but are not homotopic.

Answer 3

Without non-self-intersection, the first statement is already false for the plane minus one point, and the second statement is already false for the plane minus two points. In fact, the fundamental group of $\mathbb{R}^2$ minus $k$ points is the free group on $k$ generators (so free homotopy classes of loops correspond to conjugacy classes in the free group), and for $k = 2$ a commutator in this group (which describes the Pochhammer contour, as mentioned by Jason DeVito in the comments) is both null-homologous and has zero winding number around every point, but is not null-homotopic.