I see, the "w.l.o.g." is the important part for me right now.

@suvrit, regarding 3, that's exactly my question.

my interpretation of obvious is "your intuition and experience says something but there is no proof for that"

well, local hamiltonian is complete for QMA, but it is a promise problem. Also, 5-QSAT is complete. As Watrous puts it, "vacuous promise" which means "decision problem". So, it is not expected that a complete decision problem exists for any semantic class.

yes, I'm misusing the notation, but you completely understood my question. Thanks for the reply, know is crystal clear.

In several papers they sometimes use amplification and sometimes don't. So I was intrigue on that. It wasn't clear for me when to use it.

actually, it doesn't say that we can replace an inverse polynomial by an inverse exponential.

In Arora and Barak, theorem 7.10 page 132 it says Let $L\subseteq\{0,1\}^*$ be a language and suppose there exists a polynomial-time PTM M s.t. for every $x\in \{0,1\}^*$, $Pr[M(x)=L(x)]\geq 1/2+n^{-c}$. Then for every constant $d>0$ there exists a polynomial-time PTM M' such that for every $x\in\{0,1\}^*$, $Pr[M'(x)=L(x)]\geq 1-2^{-n^d}$. Is my interpretation correct? Of course it's not saying anything about the running time.