Basement lighting four rooms
First of all, as I edited in the question, wire normally refers to a single conductor. Multiple wires inside a outer covering is referred to as a cable. 14/2 and 14/3 are common cables, each containing multiple insulated wires (2 or 3) plus a bare ground wire.


  First question: for Room #2 from Box #2 to Box #3 - Am I correct in saying it needs 3 cables? 1st cable - Power/neutral in 14-2 2nd cable - Power/neutral out to box #3 for switch #4 and then to switch #5 3rd cable - Traveler wires (2 hots/1 neutral) 14-3 between boxes for 3 way between switches #2 &amp; #3


That is the typical way to do it, though it uses a little more wire than the minimum. You could use a single 14/4 cable between box 2 and box 3. This would have hot (black), neutral (white), 2 travelers (other colors). The advantage is you save one wire because the neutral can be shared since this is all part of one circuit. However, if you already have (or will be getting) 14/3 for other 3-way switches then running 14/2 + 14/3 is probably cheaper than getting 14/4 just for this one installation. As noted by Harper, if you use separate /2 and /3 then you must keep the neutrals matched to the hots/travelers and not connect them together in room #3 (i.e., switches #3 and #4).


  Second question: At each box would you just pigtail the power in / power out and power to the switch?


Correct. Pigtail hot (black), neutral (white), ground (bare) and connect to switches as appropriate.


  Third question: Is 14 the correct gauge wire since the whole load is less than 3 amps?


Correct. Just make sure that the circuit breaker is also 15A to match.