1. Dancing is fun.

(Is 'fun' and adverb or a noun?

In this sentence, it's an adjective, modifying "Dancing."

http://dictionary.reference.com/browse/fun

fun   /fʌn/ Show Spelled

[fuhn] Show IPA
noun, verb, funned, fun·ning, adjective
–noun
1. something that provides mirth or amusement: A picnic would be fun.
2. enjoyment or playfulness: She's full of fun.
–verb (used without object), verb (used with object)
3. Informal . joke; kid.
–adjective
4. Informal . of or pertaining to fun, especially to social fun: a fun thing to do; really a fun person.
5. Informal . whimsical; flamboyant: The fashions this year are definitely on the fun side.

(Thank you. According to the dictionary, it seems to be an adjective. Right? Or can 'fun' be an adjective and a noun?)

In the statement, "Dancing is fun," the word "fun" is used as a noun. It is the object of the verb "is." To determine whether "fun" is an adverb or a noun, you can ask yourself what "fun" is expressing in the sentence. In this case, "fun" is describing or identifying the quality of dancing, so it functions as a noun.