Can a smooth, immersed loop in R^2 become not nullhomotopic by removing a point?

You can construct an immersion $\gamma$ which remains null-homotopic after removing any point $p$ not lying in its image. It suffices to let $\gamma$ travel along a graph in such a way that it runs along every edge the same number of times in each direction. As an example, you can take a train track with one central 4-valent switch and two arcs: it looks like an "8" but the 4-valent vertex is flattened, so that each of the two circles has a cusp:

_{(source: unipi.it)}

You can let a train travel along this train track so that it runs on each of the two arcs twice in opposite directions.

For a different way of looking at the same issue see http://en.wikipedia.org/wiki/Pochhammer_contour . The contour is set up to have winding number 0 around any point.