For a start decoupling is needed, between +5 and ground and between 12V and ground.http://www.thebox.myzen.co.uk/Tutorial/De-coupling.html
The TLC5940 can operate with a 12V supply to the chip. In fact it will operate up to 17V.
I don't think I will need any resistors at all for the TLC5940.
Yes you need a current setting resistor.
Should I ground the 12V supply to the arduino?
If you don't nothing will work.http://www.thebox.myzen.co.uk/Tutorial/Power_Supplies.html
I have read that I need to get rid of the IREF 2.2 resistor and that makes sense to me because the higher voltage I am trying to use.
Makes absolutely no sense to me.
The LEDs have their own resistors already built into the strips.
That suggests they might not be suitable for driving with the TLC5940 - how much current do they take?