PaulRB:
I've been reviewing the code....
Now that in itself shows great enthusiasm! 
PaulRB:
I think the multiplexing is done by column-by-column.
In which case, no risk of damage. 
PaulRB:
I take my hat off to the coder. Knowing so little, to have battled doggedly on to get anything working at all shows impressive determination. Bravo!
Have to agree.
Now if the rows were the cathodes, then substituting the right-hand 74HC595 with a TPIC6A595 and multiplexing by row using the 1k resistors could produce a workable display.
But only as an exercise. For any serious purpose, I have advised in #9.