that wouldn't work either. A 12ax7 to 6eu7 adapter would need to feed power from 4/5 to 9, not from 4 to 5.
sort-of yes, sort-of no on the gain. A paraphase has the gain of the first triode. The second triode has gain too, but since the input is knocked down in order to get the same output voltage as the first triode it really doesn't count.
A cathodyne in and of itself is slightly under unity. What you get from grid to ground will be a wee bit more than you get from plate to ground or cathode to ground. Its usually paired with another tube as a voltage amp stage, so you still get the gain of the first tube. Overall it ends up not that different in terms of total gain than a paraphase once you figure in the two sections.
The problems with the paraphase are largely about the balance of the two sections and how it varies with frequency. On the bottom end, you get the coupling cap reducing signal to the inverter as the frequency goes lower. On the top end, you get the Miller effect in the tube causing some roll-off in the voltage divider that feeds it. If you ever set one up on the scope and balance it perfectly at 1khz, you can watch it go out of balance as you shift the frequency up and down. Tubes aren't perfect either so even if you get the balance perfect at one spot with one tube, changing tubes may shift it some.
The floating paraphase is also sometimes called a paraphase with feedback. At least thats what the RDH calls it, since thats more properly what it is. Basically by moving where the ground connection is, you get some local feedback at the tube that goes a long way towards making sure the output is actually equal no matter what the frequency is.
The Valve Wizard -Paraphase
has some more technical stuff about the two of them.
Cathodynes don't have the variable response issues, they just need matched resistors and you get balanced output. The down side being the cathode runs at a bunch of volts so H-K voltage is something that needs to be considered. On the up side, since that also means the grid needs to be a bunch positive, it makes direct coupling a possibility. Eliminating a cap between the voltage amp and inverter removes one possible phase shift point so you have less places to cause instability in the amplifier. Fisher used this with 12ax7 tubes, but you can also find the triode/pentode tubes set up this way in Dynaco and plenty of other gear.
diff pair / long tail pair amps also have gain, but its basically the same as a single triode would be. They also tend to have very good balance, though if you don't size the grid to ground cap properly it can get unbalanced at lower frequencies. Same problem with heater-cathode voltages, and also the same potential for direct coupling.
as for why you'd use one over the other, frankly I don't have an answer to that. Only things I've ever changed has been a paraphase to a floating paraphase and thats kind of like putting better tires on the car you already have. I have amps with cathodynes in both direct coupled and cap-coupled config, and one with a direct coupled long tail pair. They all sound fine but different. Wouldn't care to guess how much of the difference is the inverter and how much is the fact that nothing else is the same between them either. Too many variables to even try and figure that out.
