Unconditional authentication
$\;\;\;$ Sure. $\:$ The simplest way is to OTP-encrypt the

$\;\;\;$ output of an almost xor-universal hash family.
$\;\;\;$ That could be used for encrypt-then-MAC, where

$\;\;\;$ the MAC is applied to an ordered pair that indicates

$\;\;\;$ [the message number or how far into the pad to start] and the OTP ciphertext.

$\;\;\;$ (Presumably, the pairing function would be $\;\;\; \langle x,\hspace{-0.03 in}y\hspace{-0.03 in}\rangle \: \mapsto \; $prefixfree$(x)\hspace{.04 in}||\hspace{.04 in}y \:\:\:\:$.)
$\;\;\;$ Yes.
$\;\;\;$ The tightest way I'm aware of to get provable universality while still hopefully being

$\;\;\;$ second-preimage resistant is the trivial construction that just concatenates an output

$\;\;\;$ from a universal hash family with the output from a standard cryptographic hash function.