Modify LED circuit with a MOSFET
If you took more time thinking about drawing your circuit (extremely important) you'd be able to unclog your thoughts and see that the addition of a 1 Mohm resistor (purple) would be a good idea: -




  Note: I want the second LED strip to stay OFF even if the MCU isn't
  powered.


The 1 Mohm resistor is there to protect the MOSFET and achieve the above should the ATTiny become disconnected. It's optional of course.

It's still not drawn perfectly but is a lot better and folk might have a chance of understanding it better.


  Also is it possible for me to switch the (+) of the second LED strip
  with the MOSFET instead of switching it's (-) ground?


You could use a high-side PMOSFET and low side NMOSFET to do that like this: -



Picture from here.