Problem with powering 12V RGB strip and Arduino Uno from the same source.

A TIP31 is a bad choice.
It needs a base current of 1/10 of the collector current for saturation.
Something an Arduino pin can't provide.

A darlington, like the TIP120, would be better.
Logic level mosfets would be the best choice.

What value base current limiting resistor are you using now.
The Arduino shutting down with >9volt on V-in suggest you didn't use base resistors.
Leo..