You’re right, usually you place the verb between “ne” and “pas”, but that applies to conjugated verbs. When the verb is not conjugated, it goes after both particles, e.g. “Ne pas manger” or “Ne pas ouvrir“. Your example is not an order, but the same principle applies. “Ne pas avoir à” is not really a set phrase, but more of a set construction, but it does indeed mean “to not have to”.

Regarding your other questions, I think you mean to ask if we pronounce the liaisons between those words, as there’s not really any concept of pronouncing a phrase like it’s a single word. I think in this instance it’s optional. I wouldn’t pronounce the s, but some people would. I think it might be regional.

Native French speakers would not drop the “de”, but would drop the “ne” while speaking. This applies to a lot of sentences. And you are right about your last question. “Content” is indeed pronounced without the final t sound.