Tongji, short hop laser can be a pain, but it is not anywhere near as bad as it would seem at first glance.
The first thing you must be sure not to do is to instantly decide that it is impossible to counter some tactic. If you lock your way of thinking into the idea that something cannot be overcome, it will instantly become that much harder to beat. Mental blocks can be a complete pain, do not fall under them.
I have said it once, I shall say it again, I have not encountered anything to date that is so overwhelmingly difficult that it could not be overcome. I do not feel that anything exists that is impossible to beat either. If I cannot beat someone, I do not pass the blame onto the character. I lost because I was not as good as that guy. If I want to beat that guy next time, I have to improve.
To beat short hop laser, you must first understand the very point of it. Short hop laser is a tactic designed to give control of the battle to the Falco player. If the Falco player feels that he does not have control over you, then things will go much better for you.
Mewtwo has a few options to beat short hop laser.
First of all, you can simply shield it. This tactic works against it just fine and it works with every character. If you can power shield even decently, all the better. If you can shield off his lasers effectively, the Falco player will eventually realize that SHL will not work against you, and probably rush you. Some Falcos will short hop laser while approaching you. Sometimes, you will be able to shield grab him when he gets closer. Problem is, a good Falco will see you shielding and will grab you himself. Use your best judgement on how to react to what he does. The one problem with this tactic is that you are not moving. By not moving, the enemy can better prepare for whatever it is that he is going to do next.
Secondly, you can play edge games. Get on the edge where lasers do nothing and mess around with him there. This can be risky as his dair spike is dangerous. I would not recommend this tactic as your primary counter to SHL as it is far too risky and you have to be darn good with ledge games to make the risk worth it.
Third, you can rush him. This sounds obvious, but Mewtwo can do it incredibly well. When he is doing his SHL, you can jump way up in the air and approach him from heights that his laser cannot reach. Combine this aerial movement with a well placed teleport and you can get on top of him without being hit. If you do it very predictably and direct, you will get hit. When approaching him from the air, you can either teleport, attack, or air dodge. Use your best judgment here. If you are constantly on top of him and attacking him successfully, he will not have much opportunity to use SHL. There exists two problems with this tactic though. Number one is that the entire reason the Falco is doing SHL is to get you to approach him. With him expecting it, he will have the upper hand strategically when you get to him. Number two is that the Falco could have better close combat fighting skills than you do. Despite these short comings, I do feel that this option is the best of the three for fighting against short hop laser.
In my case, I usually do strategy one when fighting a Falco, if they are persistent and continue to do SHL, I will then do strategy three. As I am right now, I have no real problem playing against short hop laser simply due to me encountering it so many times.
Your comments on Fox vs Mewtwo with you and your friend are stated with too absolute of a mind set.. Fox can be a much more all out character as he has no truly obnoxious weaknesses.(He has some, just not on the scale that Mewtwo does.) If you run straight at the guy, you have less to worry about with Fox than you do with Mewtwo. This is not to say that Mewtwo is bad for offensive fighting; he is quite awesome at it. However, with Mewtwo, you have to attack with a certain degree of finesse. You have to have flawless timing and downright awesome spacing. If you mess up on either of those, you will be punished. The problem is, the punishment is quite severe in Mewtwo's case. If you mess up, you will probably receive a large amount of damage, and with Mewtwo, you do not need too much damage to be killed.
All in all, the best advice I can give you is to keep playing and do not give up. I have been playing Mewtwo for a very long time. I have been to countless tournaments and have played against the best that Florida has to offer. I was in the same boat as you at one point. I felt that I could not get any better than I was. I felt that nothing would change. However, it eventually all started coming together. Getting all of your tactics down and increasing your proficiency with the advanced techniques takes time. Play as much as you can with as many people as you can. Mewtwo can have terrible consequences for when you make a mistake, but he can have unbelievable rewards for when you battle with precision. A double edged sword.
Peace.