Regardless of the notation, the links in question are the (2,n) torus links (or possibly their mirror images). The knots come from the odd values n=3,5,7,9,... and the links are the even values n=2,4,6,8,.... So the answers to questions 1 and 2 are T(2,15) and T(2,10).

KnotInfo has lots of prime examples -- look at determinant and symmetry type, and anything with det(K) a multiple of 3 and amphichiral symmetry will work. The first of these is 8_18.

There's a continuous family of disks $S^1\times[0,t] \cup (D^2 x \{t\})$ with boundary $K$. At $t=0$ you have $D^2 \times \{0\}$, and at $t=1$ the whole disk lies in $S^3$.

The core of one filling is a knot $K$ in $S^3$, and then the other $S^3$ filling is a nontrivial Dehn surgery on $K$. Since $K$ has a nontrivial $S^3$ surgery, it must be the unknot (this is theorem 2 of Gordon--Luecke, "Knots are determined by their complements"), and so its complement is a solid torus.

I'm claiming this fact for any rational homology sphere $Y$ with the additional hypothesis that $\pi_1(Y)$ is "cyclically finite", a condition which ensures that the Chern-Simons functional used to define $I^\#(Y)$ is Morse-Bott at the reducible representations. (See Theorem 4.6 of that paper for the precise statement.). When $Y$ is $p/q$-surgery on a knot $K$, Boyer--Nicas proved this hypothesis equivalent to the condition that no root of $\Delta_K(t^2)$ is a $p$th root of unity.