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.