Before doing anything else, cycle the gun empty. Dry fire, hold the trigger back, and cycle the bolt. Release the trigger. Will the gun (dry) fire or is the trigger "dead"?
If the trigger is "dead" (like what happened when you tried to fire the second round), push out the rear takedown pin, and hinge the action open. Look at the hammer. Is it down? or still cocked? If the hammer is "following" the bolt when it closes, it will be down.
You can also check this with just the lower reciever. With the hammer cocked, pull and hold the trigger (easing the hammer down, so it doesn't snap against the reciever). Then recock the hammer with your thumb. It should be held cocked by the disconnector (as long as you are holding the trigger back). Let go of the trigger, and the hammer should fall slightly, and be caught by the sear nose of the trigger. If the hammer falls all the way, then either the hammer, the trigger, or the spacing of the pin holes in the lower reciever is bad.
If you have another AR lower available, put the questionable parts in it and see if they work properly. IF they do, then your problem is probably the new AR lower. If they don't, the parts themselves are at fault. Be careful not to confuse which parts go to which lower when you have them apart!
All else being equal (and it almost never is) bigger bullets tend to work better.