Would it be possible to power the detached room from the breaker box with two 30 amp breakers to the room with UF 10-2 wiring underground from the breaker box and then landed into the junction box at the room where I would join it with previously wired 12-2 wiring, one for a small 115/15 amp window unit and another for separate receptacle? Also, mentioning the fan switch and small appliance will be running on the same 30 amp breaker and the other receptacle to another 30 amp breaker. Pardon my father built this dwelling about 10 years ago and we never got around to powering the room.
Yes, but you'll have to change the breakers to 20A. You can't have 12AWG wire downline of a 30A breaker. (NEC 240.4).
And I presume you'll want to fit common 15A or 20A receptacles, and those are only allowed on 15A or 20A circuits. (with an exception allowing 15A receptacles on 20A circuits; no other substitutions allowed.) NEC 210.21(B)(3).
Another problem is you're talking about running two separate 120V circuits to an outbuilding. That is not allowed. You can only have 1 circuit of any particular voltage, unless by nature the circuit has a special requirement e.g. a lighting circuit that needs to be switched from the main building. (NEC 225.30) You can have exactly one 120V circuit, and if you want to plug anything into it, it must be 15A or 20A. (210.21b3). You can also have a 240V circuit.
The normal way you handle multiple circuits to an outbuilding is one single feeder that feeds a sub-panel. You can only have one, but the subpanel can support any number of 120V, 120/240V, and 240V circuits.
The subpanel can be any amperage. It can then have any number of 15A or 20A circuit breakers in the subpanel protecting individual branch circuits, or circuits of other sizes.