Is it okay to do certain dumbbell lifts alternating arms, or should they always be done both arms at once? With standing dumbbell shoulder presses, I can lift much more singly than both together. Is this incorrect form though?
Post a form check video. 
Single armed presses and the typical shoulder press with two dumbells are different exercises. Then there is standing vs. sitting, some use of the legs vs. a strict press, etc…
Typically single armed is easier for most people. Seated with a solid back support is easier than standing too.
Depends on the movement/exercise, but yes, perfectly OK as a rule and recommended as one method of mixing things up, etc.
Exercise is good for health and to maintain the fitness. So, along with other exercises we can also do the exercises with the dumbles, It is good for the arms or you can also visit the websites to get the more information to maintain the fitness.