Okay, assume you have OE/ low and MR high and not floating.
Am confused by this:
"the serial port receiving stops working"
How's that tied in?
You have quite a load on the '595 outputs also.
Looks like an LED with 2-3mA, and 16 transistors with ~4mA each, total of ~70mA on each IO.
Part is only rated for 5V output for 6-8mA, and 70mA for the whole part.
I could see drive issues there causing some of your problem.