The neutral and ground MUST NOT be bonded at a sub-panel. They should only be bonded at the main service panel. If you bond them anywhere other than the main service, the neutral return current now has multiple paths, including though your ground wire.

You should be able to buy a second bar for the sub-panel if it really is meant to be used as a sub-panel. The neutral bar will need to be isolated (it should have plastic insulator separating it from the case). The ground bar should be bonded to the case.

The ground wire from the house must be connected to the sub-panel ground. Since it is a detached building the ground rods are also required by code and should be tied to the sub-panel ground. The ground to the service is the most important because it will provide the low resistance path back to the service in case of a fault.

According to Table 250.122 of the National Electrical Code, a 10 AWG copper conductor is fine as an equipment grounding conductor for circuits with up to 60 ampere protection.

National Electrical Code 2014

Chapter 2 Wiring and Protection

Article 250 Grounding and Bonding

250.122 Size of Equipment Grounding Conductors.

(A) General. Copper, aluminum, or copper-clad aluminum equipment grounding conductors of the wire type shall not be smaller than shown in Table 250.122, but in no case shall they be required to be larger than the circuit conductors supplying the equipment. Where a cable tray, a raceway, or a cable armor or sheath is used as the equipment grounding conductor, as provided in 250.118 and 250.134(A), it shall comply with 250.4(A)(5) or (B)(4). Equipment grounding conductors shall be permitted to be sectioned within a multiconductor cable, provided the combined circular mil area complies with Table 250.122.