Can I wire 2 ceiling fans, 5 ceiling spot lights, and 3 closet lights on a singe 15 amp circuit?
There should be no problem putting it all on a single 15 ampere circuit. Likely in the worst case scenario, each fixture would consume ~100 watts. That would be using large 52" fans, and 100 watt bulbs in each fixture. With the 10 fixtures, that would be 1000 watts (10 * 100 = 1000). A 15 ampere 120 volt circuit can provide 1800 watts of power (15A * 120V). 

More realistically, you're going to be using lower wattage fixtures, especially nowadays with CFL and LED bulbs becoming more common. Here's some calculations based on various wattage devices. 


10 devices @ 75 watts each = 750 watts.
10 devices @ 60 watts each = 600 watts.
2 fans @ 100 watts, and 8 lights @ 60 watts = 680 watts.
2 fans @ 75 watts with 3 60 watt blub light kits, and 8 lights @ 100 watts = 1310 watts.
2 fans @ 100 watts with 3 100 watt bulb light kits, and 8 lights @ 100 watts = 1600 watts.


Since it's a 15 ampere circuit, you'll only need 14 AWG conductors. Unless there's a specific reason you're using 12 AWG instead, which is also fine.