The first step is to figure out exactly what you need for the antennas you want to put up.
If you are trying to match a doublet fed with twinlead using a tuner designed for coax, then
you would be looking for a balun to do that. Probably the simplest type is a length of coax
wound through a ferrite core to make a 1 : 1 current balun. (While you could use a 4 : 1
balun, the general consensus is that the 1 : 1 balun is usually a better choice if you don't
know the specific impedances you have to match.)
G3TXQ's page gives lots of options: http://www.karinya.net/g3txq/chokes/
For a vertical or end-fed wire, just attach it directly to the tuner. (You can get adaptors from
a BNC connector to a pair of binding posts that will make it easier to connect the wire and
radials if your tuner just has a coaxial output jack.)
You don't need an un-un: just connect the tuner directly to the wire. Same with the
telescoping whip, though you may find that several shorter wires are more efficient
as radials than just the two longer ones.
If your tuner can't match the whip on 40m or 80m, then you may have to make a loading
coil for the base to improve efficiency. Generally, however, you'd be better off using one
of the 40' wires tossed over a tree branch for a radiator rather than the much shorter whip.
AD5X's matching transformer is for low impedance mobile antennas fed with coax: as long
as you can mount the tuner right at the antenna, the tuner should have sufficient matching
range without it.