Can I install this array of circuit breakers in my sub-panel?
Short answer: yes for the branch circuits, no for the feeder. 

The capacity of the sub panel is a total of 30 amps of load NOT the sum of the breakers. As long as the total of all of your equipment doesn't exceed that then you are good.

Edit: Tester is right and I missed that you had 12/3 running to the shed. #12 wire cannot be protected at 30 amps. The maximum for #12 is 20 amps. If you want a 30 amp circuit to the shed it needs #10 wire.