For what it's worth, two of your 3 points relate directly to the mental aspect of baseball. And your 3rd point (which is actually your second one as listed), may, in fact, be mental. If you start wondering and worrying if its the bat, the problem becomes a mental one. I don't personally think the bat is an issue just based on what you wrote.
Get the proper swing down and then TRUST IT. Go up there believing you will hit the ball. Yet realize that if you don't this particular at-bat, you will get the next one. By that I mean don't dwell on failures, because it only heightens the worry response.
Understand it's ok to be nervous / anxious. That's where trusting the swing you practiced over and over comes into play. You know you have the write swing mechanics, so even though your a bit anxious, you can still trust your swing and know that you'll get the desired results. But you must first be confident in your swing mechanics. Do visualization exercises where you see yourself getting solid hits. Video tape an at-bat where you hit the ball well, and just watch it over and over to burn into your subconscious the proper swing mechanics with the expected results.
Post a followup: