I also had a really tough time cramming three fingers down on the 2nd fret for an A maj chord. The tip of my middle finger is a bit big and fat compared to the others.
I learned it was a lot easier for me to just use the index and middle finger to hold down and fret all three strings. Alternately, you can try the middle and ring fingers. See if that helps.

As for your guitar, an action that's too high can cause problems especially with beginners. If you know someone familiar with guitars they can look at for you and judge if they think there's a problem with the action. You can also measure it yourself if you have a decent ruler that goes down to 64ths or 32nds. But try some alternate fingerings for the A chord to see how it goes.