So if 1 LED is around 20ma, I can drive up to 7 LEDs per output of the tpic6b595, right?
More than that. In your diagram above, you show 2 parallel groups each of 6 LEDs in series. I think 6 is probably too many. 5 might be ok. You can do this because your supply voltage, which could be as low as 12V, is still many times higher than the forward voltage of a single red led, which is normally around 2V. The total forward voltage of 5 LEDs is 10V, leaving 2V to be dropped by the series resistor (eg. 100R). The important point is that the LEDs in series are not drawing 5 X 20mA, the same 20mA flows through all 5 LEDs. That means a single output of the tpic chip could support 7 parallel groups of 5 LEDs in series, 35 LEDs in total.
, do you think the 74hc595 will be fine for my project?
No, the max current for 74hc595 is 70mA total for 8 outputs and the max voltage is 5 or 6V, so not suitable at all.