Does Warpath/Acid Storm prevent the initial two battle flips for defense/attack respectively? In other words, are the initial two battle flips considered bold/tough?

No.

I attack into my opponents Acid Storm which has 0 defense. Before attacking I play Press the Advantage. My opponent flips defense cards and gets 3 defense. Does Press the Advantage bring him down to 1 defense or because it was played when he had 0 defense does he stay at 0 because the card says he can't go below 0?

In this scenario, Press the Advantage would bring your opponent’s character down to 1 defense. Any attack or defense modifiers apply for their stated duration. Only when the result is less than 0 do you use 0 instead.