Subtraction of Power Series
OK I thought maybe you were confused about how subtraction works as opposed to addition.
If we're adding two power series, as long as both power series converge their sum converges (you should be able to prove this using that limits split into sums). So as long as both of your power series converge, the new one will - hence the radius of convergence is AT LEAST the minimum of the radius of convergence of the orginal two series
For the second question, are you saying it converges on every compact subset of the disk? If so then clearly the radius of convergence is at least 1