Depends on how the threaded part is stuck.

Traditionally, you can back it out by using a pointed object (e.g. nail or screwdriver) then tapping on it so that it rotates counterclockwise.

Some have had success by simply putting a strong adhesive (e.g. Mighty Bond) on the portion that broke and twisting the assembly out after the adhesive has cured.

Yes, there are non-OEM stubby (short) antennas available in the aftermarket. But you have to go through trial and error to find one with the same thread as the Picantos. Iba iba din kasi depende sa brand.