Gravity Falls s1e4
Aww, this is like a bizzaro version of a Mystery shack. "Bizzaro version" - what does it mean?
It ain't some such old trick, it's an natability. "natability" - ?
Where others are blind, I'm futurely inclined. "futurely inclined" - ?
And you to could see the view that's widdle lil me! "widdle lil" - ?
I succesfuly bezappled my face! "bezappled" - ?
little scug's house. "scug" - ?
some digs you got here. "digs" - ?
geopardy quicksand! "geopardy" - ?
hiertown. "hier" - ?

Question

Gravity Falls s1e4
Aww, this is like a bizzaro version of a Mystery shack. "Bizzaro version" - what does it mean?
It ain't some such old trick, it's an natability. "natability" - ?
Where others are blind, I'm futurely inclined. "futurely inclined" - ?
And you to could see the view that's widdle lil me! "widdle lil" - ?
I succesfuly bezappled my face! "bezappled" - ? 
little scug's house. "scug" - ?
some digs you got here. "digs" - ?
geopardy quicksand! "geopardy" - ?
hiertown. "hier" - ?

Answer

"Some (nice)/ (sweet) digs you got here" --Digs is slang for "place", like where you live. 
The sentence is a compliment to the host. 
other slang for place: digs, crib, casa, pad, abode, lair

I would say to ignore the other phrases on your list; they're obscure and uncommon. If you like learning slang, go to www.urbandictionary.com

Answer

These sound like evocative song lyrics in which the songwriter is playing with the sounds of words, and making up new words and phrases.

"Bizarro" is a reference to the Superman comic series. See http://en.wikipedia.org/wiki/Bizarro . Bizarro was a flawed copy of Superman. A "bizarro version" means a flawed, warped, or distorted version.

Answer

These are hardly common expressions, some are slangy and a couple seem invented for the show. Some phrases are deliberately "bad English". It's probably better if you find a fan of the cartoon and discuss the meaning with him/her.

I can tell you that Bizzarro is the "anti-Superman" in DC Comics. So that analogy is applied to the Mystery shack.