Morph was a creation of the cartoon that wasn't actually in the comics. They later said that a character from long ago called the Changeling was who Morph was in the comics but he died back in, like, the 60s or 70s. A very minor character.
There was later a storyline called the Age of Apocalypse where thy created a Morph character (Changling who lived) who looked nothing like he did in the cartoon or old comics. Here he is:
The Age of Apocalypse was an alternate world created when Xavier's son went back into the past and accidentally killed Xavier before he formed the X-Men. Jump forward in time and we see what life would have been like without Xavier, and Morph is one of the X-Men, led by Magneto. Oh, and Apocalypse rules the world, which is pretty much a nightmare. I highly recommend getting the trades. Great run!
Anyhow, after that, there was another Morph created (just like the Age of Apocalypse Morph) and he joined a team called the Exiles. He was plucked from another timeline. The Exiles jumped from one world to the next, fixing problems to save the universe and Morph was a mainstray on that team for years. The series was fantastic for the first 50 issues or so. Sadly, since that series ended he's not really shown up anywhere.
So as of now, there is no Morph anywhere.
Oh, and my opinion is if you want to follow X-Men (as I am a hardcore X-Men fan) you should read Uncanny X-Men and All New X-Men for the main X-Men stuff. They're the core titles. Wolverine & the X-Men isn't quite as important but still enjoyable. There's also a new book, "X-Men" coming out very soon staring a team of X-Women that looks really good. I'd recommend that.
I wouldn't recommend Astonishing X-Men or X-Men Legacy. They aren't that great or important overall, but if you want to try them then there's no harm in that.
Outside of the core X-Men book, Cable & X-Force sounds like something you'd enjoy. Uncanny X-Force is just alright so far. I don't know if I'd recommend it. X-Factor has characters from the comics you mentioned like Shatterstar, Rictor, and Monet (from Generation X). I get bored with it but most people like it. It might be worth picking up. Uncanny Avengers is a team of Avengers and X-Men mixed (the X-Men being Wolverine, Rogue, and Cyclops' brother Havoc... the Avengers being Captain America, Thor, and Scarlet Witch) and they're taking on threats from both Avengers and X-Men worlds. They just finished up an arc with the Red Skull and next will be Apocalypse. It's been delayed badly due to the artist so it's only on issue 4 but they're getting a new artist so it should be coming out more regularly soon. It's been decent, worth the pick up.