Yes, you can power the strip from a different source and use the Arduino on the data pin only - but you must remember to use a common ground.
Please do not add delay where you did, it will not do anything else than slowing down you code for no avail.
If you need animation in the strip / colors, you need to make that animation. Moving the "set all red" out of the main loop may be contra-productive if it is part of an animation on the strip.
Low power mode is of course useless if you want to run an animation on the strip. My suggestion was aimed at the fact that you only wanted the strip to be red.