Let $T$ be a locally unknotted $2$-tangle in $B^3$ and $\Sigma(T)$ be its double branched cover. Can $H_1(\Sigma(T))$ have a non-trivial torsion? (Obviously, not for rational tangles.)
Studying this torsion has a nice application due to David Krebes (An obstruction to embedding 4-tangles in links, J. Knot Theory Ramif.,8 (1999), 321-352.) You can use it to show that a given 2-strand tangle doesn't embed as a subtangle of the unknot.
An example (similar to Ken's and related to Ian's comment): take the pretzel tangle P(-3,-3,3) drawn below; its double branched cover has homology $Z \oplus Z/3 \oplus Z/3$. This is from my paper (Embedding tangles in links, J. Knot Theory Ramif.,9, No. 4 (2000), 523-530).
Yes. For example, start with the exterior of the Whitehead link and do p/q filling on one component where p is odd. This manifold has p torsion and is the double branched cover of a two strand tangle in a ball.
You can see this by taking the quotient of the Whitehead link by the strong inversion and doing the appropriate rational tangle replacement on one component. The exterior of the other component (in the quotient) is then a two-strand tangle in the ball.
Note that if p were even, then you'd also have a single closed component in the tangle in addition to the two strands.