I personally don't have much difficulty fighting with Links. Shadow balls work wonders for countering Link's projectile spam. About it being a matchup in Mewtwo's favor, I am not so sure, but it certainly isn't one of Mewtwo's harder matchups.
Ice Climber fighting is a bit tricky, but can be dealt with. Neutral air is excellent for separating them. Down tilts are good for stopping a fast IC that is trying to come in and dsmash or grab you. They(down tilts) are also effective at starting the trusty down tilt to fair combo, which works quite well on ICs.
Throws take some care to do correctly against an IC. If you grab one, a good IC player will have the other one attack you whenever possible. Learn how to throw intelligently against them. Suppose I grab one of the ics. I am holding it right in front of me. The other ic can run forward and smack me. If I push back on the control stick, I begin moving the IC over my shoulder for a back throw. The instant the grabbed IC starts moving, it becomes a weapon. If the other IC tries to save the one I have caught in the grab and I initiate the grab at the right time, it will both smack the non grabbed IC and throw back the other one. Forward throw can work well as the shadowballs being shot at the grabbed IC will usually hit the non grabbed one. I would recommend against up throw unless the ICs are separated and the other one will not be able to get to you until the throw is completed. Down throw is useful as it is a very quick grab. You can often pull it off without punishment.
Something I would say is important for any and all matchups is to learn to get out of grabs quickly. ICs grabs are dangerous, especially if your opponent knows how to wobble and it isn't banned where you are playing at. When an IC throws you, DIing up is generally the way to go.
Keep your spacing and don't get nervous. A good IC player can be incredibly fast. This can cause some players to get overwhelmed and become defeated psychologically as a result. As long as you keep your cool, you should be able to do fine.