Using just shift registers with resistors will work, but it won't be very bright. Looking at the shortage of time, it may be the way to go. Wiring up the matrix will be unpleasant enough. Remember: one resistor for each color per led per row/column.
If you plan to use the pnp transistors and you have a decent electronics store in your town that is open on Saturdays, you should try getting common anode leds. You could ask for 8ch led drivers as well, but that might not be successful. Then you would be done with just 5 pnp transistors for the 5 rows and could use 3 shift registers for the 15 cathodes (and resistors, 20mA max.) (per row. but they are shared). That would be a bit brighter than just using shift registers for sourcing and sinking current. But be warned, that you cannot use anything else than 5V for the leds as well. If you need a higher supply voltage, you'll need npn-s to drive the pnp-s, otherwise they never turn off. The arduino would have to output more than 5V, which it can't.
It appears that you have built your circuit on breadboard. If you get strange behaviour, make sure that the wire's actually have good contact and are thick enough to make that happen. This is one short path to insanity.
Substituting the 10Ohm resistor with a wire is not a good idea. It would help if you made another drawing indicating where you put your volt-meter's leads.
But before you do that:
I bet 'Grumpy Mike' would probably use severe language for the schematic you've posted. With just one resistor per led it just can't work, you need 3 per led. It should be trashed. Also it doesn't make any sense to share the transistors across the leds, if you need individual color control (with the exception of shades of gray for all of them).
And regarding your other question: yes it is me. There can only be one
Now I must go and travel to some of the still remaining parallel universes and eliminate the duplicates.