Can a grounding rod be connected to service neutral only

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.

Yes, this rod is correct, only it is NOT "additional", it is required since this feeder runs to a detached structure.

Also, as mentioned, many/most inspectors will as if the ground rod meet the 25 ohm requirement. If it does not then a second rod is required and the 25 ohm rule is no longer in place.

Also, and this is a biggie, that SOOW cord is absolutely NOT code legal in this case. Rubber cord CANNOT be used for permanent wiring. You need to use either individual THHN/THWN conductors if the conduit is complete from panel to panel, or you can use UF cable as long as the size cable you get can legally fit in the conduit. Individual conductors in conduit could be #8cu, but due to the distance and the small cost difference I'd go with #6, actually #6 for the hots, #8 for the neutral, and #10 for the ground. Uf cable would need to be 6/3.

I would seriously consider making this right and getting the whole job inspected.