Hi there,
Short answers: Yes, Sort of, and Yes.
Longer answers:
1) No problem soldering to the leads of the wall wart.
2) I say 'sort of' because in the diagram you indicate that you want to use the resistors to drop the 12VDC to ~3.5VDC--but LEDs are current controlled, not voltage controlled. You want to use the resistors to limit the current they will draw, since (unlike an incandescent bulb) an LED will draw as much current as it can, and if that means it draws enough to burn out or even pop, it will happily do it.
So instead of thinking in terms of the voltage you provide to the LEDs, think in terms of the current. The formula to determine the resistors you will need is R = (Vs - Vled) / I, where R is the value of the resistor in Ohms, Vs is the voltage of the power supply, Vled is the forward voltage of the LED, and I is the current you want to supply.
Do you know the forward voltage and current rating of the LEDs you intend to use? How many LEDs do you want to use? I'm guessing more than four, but that's just a guess.
There is a useful tool for calculating the resistors you need for strings of LEDs in different configurations here: **broken link removed**
There is some good information for basic LED work here:
Light Emitting Diodes (LEDs)
Good luck!
Torben