Welcome.
The PS3 controller isn't like other controllers, and it can't be added to some existing arcade stick layout or another controller without doing some more work, so make sure this is going to be a PS3 only stick and that none of the switches in it are already wired together in any way. If you're making one from the ground up then you're good to go. You'll be able to wire up the COM lines with only 1 wire on the board for each one, and then just jumper around to the other buttons you need, instead of 1 COM wire for each button run back to the board (there's no room for that anyway) but they don't all share the same COM line. All of that info is in this thread also.
You can use any value from 2k to 10k really, though I'd just go with 10k as they are easier to get here and there. You will only need 2 of them for this.
On that VX5 there you can install the first Resistor from TP10 to TP17, then the other one from TP10 to TP18, and that will take care of the Pull-Ups that were on the Daughter board, so it can be tossed.
I don't recommend it for additional buttons and such, but since you're building an arcade stick from the controller, the best way to solder up all of the wires to where you need is to use the existing connection where the Daughter board mated up to the controller board, all of those black rectangle looking spots on the top of the board. You'll need to carefully scrape off that carbon coating and expose the copper, then tin it up with some new solder and then you can use those spots to solder up your wires. They're still small, and I'd use a 30awg size wire, but they're far easier to solder to than trying to prep and solder to the Vias on the board.