Will this schematic work - basic LEDs + strips
A couple things to touch on here. 


first, your battery life calculation. Stacking batteries in parallel adds current (and current capacity). Stacking in series to get 12V like you've done will not increase capacity beyond 2Ah. If your draw is about 600mA, that'd only be 3 or 4 hours
next, I would try to put 3 LEDs in series where possible so you only have to drop ~2 or 3 volts over the resistor.  150 is the correct resistor for 3 LEDs if you want 20mA per group of 3, though if you stay with the group of 2, that resistance should be higher (300 or 330)
you're correct, no external resistor needed for an LED strip (almost always). 


Other than that I think this would work just fine. Make sure that switch (and batterry!) is rated for whatever the total current draw is. Not sure what the current draw is for your length but I'd check it out.

Also your schematic has the battery flipped upside down, just a heads up!