With regard to the program executing when the clock signal is disconnected:

CMOS circuits require that all inputs be connected to power, ground or
a signal.  Otherwise the input can float about at unpredictable levels
causing intermittent and spurious behavior.  We sometimes troubleshoot
circuits by touching the CMOS chips to see if body capacitance will
cause a change of state when things won't run properly.  I suspect you
are actually providing a crude clock signal to the PIC with the clock
input floating about at strange levels and being changed enough as you
fiddle with the wiring to cause a clock signal for the program.

I don't have any experience with PIC devices (I'm expecting delivery of
my PICstart programmer today) yet so I find this list very informative
when people are dealing with these kinds of issues.

Good luck,

Andrew