Not all Strat pups have the triangular base but... i cut mine off with a craft knife and tucked the wires underneath. The triangular piece serves no purpose really.
Not sure about the parallel business, thats your personal choice. I fitted noiseless DiMarzios (YJM and HS3).
Did you have to reattach the wires to the pickup, or did you just basically cut the part off while leaving the wires attached and rig it underneath?
Also, what type of screws will I need to mount the singles?