Q1 If you are proving that a problem is NP-complete, you don't have to
prove the correctness of the reduction.

Q1 For questions about graphs remember that G=(V,E) doesn't have to be
connected, i.e. a graph might have several components.

Q3 To show that the set of decidable languages is closed under
complements, you have to show that if L is any decidable language,
then so is the complement of L.  Similarly, to show that the set of
decidable languages is closed under union, you have to show that if L1
and L2 are any two decidable languages, then their union is also
decidable. 

Q3 Lexicographic order is analogous to alphabetic order. For example,
if your alphabet sigma is {a,b,c}, then the set of all strings over
sigma (i.e. sigma*) in lexicographic order is given as follows:

a,b,c,aa,ab,ac,ba,bb,bc,ca,cb,cc,aaa,aab,aac,aba,abb,...

so if an enumerator enumerates a language in lexicographic order it
enumerates all the members of the language in the above order,
skipping those strings which are not in the language.