First attempt at looping through (not full) matrix led strip...

Yes, the wiring is different to use the ATMega SPI hardware.

What you haven't done is set up any of the registers:

digitalWrite (10,LOW);
SPI.transfer (insert_register_name_ADDRESS); //
SPI.transfer (0x0C); // and value
digitalWrite (10, HIGH);

for these registers:
DECODE_MODE 0x09 // write data 0xFF, Code B Decode for all digits << You will set this for No Decode mode instead - FIND VALUE IN THE DATASHEET
INTENSITY_ADDRESS 0x0A // 0x07 to start, half intensity. valid from 0x00 (min) to 0x0F (max)
SCANLIMIT_ADDRESS 0x0B // 0xFF, all 8 digits on
SHUTDOWN_ADDRESS 0x0C // 0x01, normal operation (0x01 = shutdown) - powers up in shutdown mode
DISPLAYTEST_ADDRESS 0x0F // 0x01 = all lights on full, 0x00 = normal ops

and then write the data you actually want to appear in these registers, which controls what appears at the LEDs (outside of writing a 1 to DISPLAYTEST_ADDRESS which turns them all on for a test):

register1 0x01 // digit 0
register2 0x02 // digit 1
register3 0x03 // digit 2
register4 0x04 // digit 3
register5 0x05 // digit 4
register6 0x06 // digit 5
register7 0x07 // digit 6
register8 0x08 // digit 7

So put the register in an array so you can reference them, and make an array to hold your display pattern

byte register [ ] = {0,1,2,3,4,5,6,7};
byte registerdata [ ] = {B00000001,B00000010, B00000100, B00001000, B00010000, B00100000, B01000000, B10000000}; // initialize with some data

Next, have a group of these in void setup to set up the control registers - each register!

digitalWrite (10,LOW);
SPI.transfer (insert_register_name_ADDRESS); //
SPI.transfer (0x0C); // and value
digitalWrite (10, HIGH);

and in void loop, write the data from an array that contains the LED pattern you want to see:

void loop(){
if (display_update == 1){ //see if "decision code" has made changes to the array
// clear flag for next pass
display_update = 0;
// now do the transfers to update the '7221 registers
for (x=0, x<8; x=x+1){
digitalWrite (10,LOW);
SPI.transfer (register[x]); // register to write to
SPI.transfer (registerdata[x]); // and value
digitalWrite (10, HIGH);
}
// "decision code"
// change registerdata[ ] based on elapsed time, buttons pressed, predefined sequence, whatever
:
:
// when all done set the update flag:
display_update = 1;
}