The Wikipedia article on Ubiquitin gives a pretty good answer to your questions. Look at the referenced articles if you want to get more detailed answers.
Are they just always available for the Ub to find to during the ubiquitination process?
Yes, This [Ubiquitination] process most commonly binds the last amino acid of ubiquitin (glycine 76) to a lysine residue on the substrate.
Is there a free lysine available on each of the additional Ub added to the chain?
Ubiquitin has seven lysines and each one of them can be bound to the N-terminal glycine of the next ubiquitin.
After the first Ub is added onto the target protein, does the E2/E3 complex still have to bind each time a Ub is added to the chain?
Yes, but this is more complicated than the general mechanism. A poly-ubiquitin chain can be attached just like a single ubiquitin or one more ubiquitins can be conjuated to an exisiting ubiquitin-protein complex. See for example Brown et al and David et al.