DC electromagnet continuous operation
If  a 5 Ohm solenoid is connected to a 1.5 V AA battery, the resulting current will be 300 mA. 


Correct.


  For a 1000 mAh battery this should provide about 3.3 hours of operation. 


It probably won't. You need to read the datasheet and see at what discharge rate the 1000 mAh figure is rated. E.g., It may be at C/10 (100 mA) or C/20 (50 mA). The mAh rating will be reduced at higher discharge rates.


  Is it practically possible to connect a 5 Ohm solenoid to a stabilized DC power adapter with 1.5 V output? 


Sure, but why? The solenoid will generally be tolerant of an unregulated supply such as full-wave rectified, un-smoothed DC. There may be no need for regulation unless the supply voltage varies too much.


  Will the voltage still be 1.5 V with such low resistance load?


Yes, provided the power-supply current rating is greater than 300 mA.