does it tend to "press itself" and stick down after you push it, or randomly at any time?
It seems that only the down button presses itself randomly, but it doesn't stay down for more than a single press. I think that the up button may occaisonally press itself as well, but only after I press down. For example I can press down on a menu and it responds but then almost immediately the up button presses itself.
My Happ Super joysticks tend to bounce back and press the opposite direction if I push them one direction and release. It's probably just that.
does it stay pressed down while you're pressing it, or does it also have the random release/repress problem?
It has a random release/repress problem for all buttons including directionals.
A random release/repress problem on all buttons is consistent with a ground loop having a shaky connection... but since you say it works fine connected to the playstation itself, I'm not sure if that's the case. If you know anybody else with the MadCatz controller you have, maybe you could borrow it and see if the button problem happens without the parts you hacked onto it.
It's also possible the controller might draw more power than the converter can provide. If you have a voltmeter, one way to test this would be to check the voltage on a few of the pins of the plug. If you hold the controller's plug horizontally in front of you, with the rounded part facing down and the pins of the plug facing you, then call the leftmost pin number 1 and the rightmost pin number 9. If you can find a way, measure voltage between pins 4 and 5, and pins 4 and 3, with the gamepad plugged into your playstation, and with it plugged into your converter. The voltage between 4 and 3 should be 7.6v, and the voltage between 4 and 5 should be 3.5v. They might be off by 0.1v or 0.2v and that's normal. If the voltages are significantly lower when plugged into your converter than when plugged into your playstation, that means the converter can't supply enough power. If power is the problem, you might be able to build something to boost the supply and make it work.
do they have the random release/repress problem when you hold down any other buttons? or only the down button? try holding down more than two buttons, without pressing down.
The release/repress problem occurs with all of the buttons. Holding down multiple buttons presents the same problem.
I'm thinking that since holding multiple buttons down probably draws more power than holding a single one down, it probably points to power supply. But, it might still be an incompatibility too. From what I've read of the playstation controller communication protocol, it just seems the kind of thing it would be hard to be incompatible with and still function with any playstation system.
Good luck with the voltage testing, if you're going to bother. I suggest using the beeping continuity tester on a voltmeter, sticking one contact in a controller pin, and touching the insides of the controller where the wires connect to the board. The one that beeps will be the wire that connects to the plug pin you have the other lead stuck into. Or you can measure resistance and look for zero ohms. Once you've figured out which is which, it'll be easy to test the voltage with it plugged in. Just be careful not to accidentally short anything while you're testing.

Here's a web page I found about the pins on a playstation controller:
http://www.geocities.com/deonvdw/Docs/Diagrams/Playstation/PSXLinux.htm