I am asking for some sort of generalization to Perron's criterion which is not dependent on the index of the "large" coefficient. (the criterion says that for a polynomial $x^n+\sum\_{k=0}^{n-1} a\_kx^k\in \mathbb{Z}[x]$ if the condition $|a\_{n-1}|>1+|a\_0|+\cdots+|a\_{n-2}|$ and $a\_0\neq 0$ holds then it is irreducible.) 


This would answer a second question about the existence of n+1-tuples $(a\_0,\dots,a\_n)$ of integers for which $\sum\_{k=0}^n a\_{\sigma(k)}x^k$ is always irreducible for any permutation $\sigma$.
What happens if we restrict $|a\_i| \le O(n)$ ? $ |a\_i| \le O(\log n)$ ?



Is a polynomial with 1 very large coefficient irreducible?
OK, about your second question. Let's consider the polynomial $x^n+2(x^{n-1}+\ldots+x^2+x)+4$. I claim that this polynomial is almost good for your purposes: if we permute all coefficients except for the leading one, it remains irreducible. Proof: if the constant term becomes 2 after the permutation, use Eisenstein, if not, look at the Newton polygon of this polynomial mod 2 - you can observe that if it is irreducible, it has to have a linear factor, which is easily impossible. 


[If we allow to permute all coefficients, I would expect that something like $9x^n+6(x^{n-1}+\ldots+x^2+x)+4$ would work for nearly the same reasons...]


