Learn French: Nous discutons en faisant la vaisselle.

en faisant is the French gerund form. It is a very natural and compact way to express that two actions are happening at the exact same time and are being done by the same subject. Here, it translates perfectly to 'while doing'.

How exactly is the word faisant formed from the verb faire?

To make this form, you take the present tense nous form of the verb, which is nous faisons. You drop the -ons ending to get the stem fais-, and then add -ant to get faisant.

Does discutons mean that we are having a serious discussion or an argument?

Usually no. The verb discuter is a bit of a false friend. While it can mean to debate or discuss, in everyday French it is mostly used simply to mean 'to chat' or 'to have a conversation'.

What exactly does la vaisselle mean in this context?

la vaisselle is a singular noun that refers collectively to all the dirty plates, glasses, and silverware used during a meal. The set phrase faire la vaisselle is the standard way to say 'to do the dishes' or 'to wash up'.