What would the total current requirements be for these items? I'm using 5 of those 8mm NeoPixels + a 5V Trinket.

I'm trying to get a rough estimate of the current requirements that would have, assuming all 5 NeoPixels were running at full brightness "white" (255 RGB).

I'm voltage-regulated to about 150 mA from the Trinket and my current hunch is that those NeoPixels need more than that, but trying to confirm it.
According to the data sheet each LED takes a maximum of 3*20mA, so you are pulling 300mA.  You  are going to need a bigger power supply :(