There is no "n" pronounced in this French word. The /a/ vowel is nasalised, instead. The sound is like the vowel in "man" and not the vowel in "top". There is a "y" sound (/j/) at the start of the vowel. Trying to write spellings without IPA is always open to misunderstanding, so we might mean exactly the same thing.
Sorry, I do not agree. In general liaison is not done after a verb or a noun.
The Z liaison after "avons" is optional, but welcome.
Certain endings typically indicate the gender. There's a simplified as well as an elaborated list on this webpage that can help you with the same: https://frenchtogether.com/french-nouns-gender/#A_simplified_list_of_endings
Of course, there are exceptions. But that's the case with any language.
Some verbs, like "have" are things that happen for a while (be, hope, know) - they don't just happen and then stop. These are called "stative verbs".
We don't use stative verbs so much in the "are ____ing" form. Compare how natural these sound:
We are happy vs. we are being happy
We know that vs. we are knowing that
The "are having" form is used for meals, though, so "we are having a dog" could be the answer to the question "What's the roast tonight?" It's unlikely to be the answer to the question "What pets do you have?"
Hope that helps!