Immersed quasi-Fuchsian surfaces surviving Dehn fillings
Notice that Hatcher proves that three-manifolds with a single torus boundary have only finitely many embedded boundary slopes. So I assume that your question one is asking about "immersed boundary slopes": the slopes occurring as boundaries of essential immersed surfaces. You ask if knot complements have only finitely many such. The answer is "no" for two-bridge knots. See the paper "Virtually embedded boundary slopes" by Joseph Maher (http://arxiv.org/abs/math/9901041). (NB - Maher uses boundary slope as a synonym for "slope".)
Regarding question two: I don't have a reference or proof at hand, but I'll boldly guess that all once-cusped hyperbolic three-manifolds have infinitely many immersed boundary slopes.
Regarding question three: Again, I think your order of quantifiers is strange here. It doesn't matter how "far" the immersed boundary slope is from the filling slope, if the filling slope is, say, the meridian slope for the knot complement.
For question 2., one can prove that for a 1-cusped orientable hyperbolic 3-manifold $M$, all but finitely many slopes bound an immersed essential surface. This follows by combining the hyperbolic Dehn filling theorem (in fact, the $2\pi$ or 6 theorems suffice) and Kahn-Markovic's theorem. If the slope $\alpha$ has length $> 6$ in an embedded cusp, then the core of the Dehn filling $\gamma$ will be infinite order and the filling $M(\alpha)$ will be hyperbolic. By Kahn-Markovic, there is an immersed $\pi_1-$injective closed orientable surface $f:\Sigma \to M(\alpha)$. Assume that $f\pitchfork\gamma$, then pass to the covering space $ \pi: N\to M(\alpha)$ so that $f$ lifts to a map $\tilde{f}:\Sigma \to N$ which is a homotopy equivalence and an embedding. Then $\tilde{f} \cap \pi^{-1}(\gamma)$ consists of finitely many points. If $\tilde{f}\cap N-\pi^{-1}(\gamma)$ is not incompressible in $N-\pi^{-1}(\gamma)$, then one may perform compressions via an isotopy in $N$ (since $N$ is irreducible and $\tilde{f}$ incompressible) reducing the number of intersections until $\tilde{f}\cap N-\pi^{-1}(\gamma)$ is incompressible in the complement of $N-\pi^{-1}(\gamma)$. Then this surface will push down to $M$ to give a surface with immersed boundary slope $\alpha$. One might object that the surface might be homotoped in $M(\alpha)$ to completely miss $\gamma$. But one may use properties of the Kahn-Markovic surfaces and the fact that $\gamma$ is essential in $M(\alpha)$ to show that this doesn't happen if the surface is close enough to being geodesic (via an argument of Bergeron-Wise).
The more recent paper of Cooper-Futer proves the existence of closed quasi-fuchsian surfaces in $M$ using the Kahn-Markovic surfaces of Dehn fillings as a starting point (so similar to this argument).
Baker and Cooper arXiv:math/0507004
MR2417445 (2009i:57035)
A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds.
J. Topol. 1 (2008), no. 3, 603–642.
In the final sections of the paper, ... (3) showing that if the interior of a compact manifold with torus boundary admits a hyperbolic metric, then every slope on the boundary torus is a multiple immersed boundary slope (MIBS).
This means there is an essential immersed surface with two boundary components each of which is a power of $\gamma$.