Leaded Ferrite bead good choice for winding my own coils?
Ferrite beads are typically designed to be lossy, to absorb energy that would otherwise be conducted or radiated as EMI.  Therefore if you want a high Q inductor they are a bad choice for a core.

Also, wrapping wire around the core the way you describe (if I understand correctly) would increase your inductance but soldering the ends to the wire through the bead would short out your coil. The core is not intended to be used in this way and you would be way better off getting some ferrite cores intended for the application you are looking at.

The two leads are a part of the same conductor, usually its just a conductor through the core or sometimes with 1 or 2 turns through a core.