Question 1
Why is it notre vie but not nos vies? Does notre vie imply that we are living together or cooperating, while nos vies imply that we have separate lives?
Question 2
Consider the sentences Il se brosse les dents and Il a mal à l'estomac. Here we do not say ses dents or son estomac as it is redundant. Then why do we say notre vie but not la vie?
Whether you say nous gagnons bien nos vies or nous gagnons bien notre vie is often context dependent, not always.
Nous sommes six associés, une quinzaine de salariés. La demande est forte, nous gagnons bien notre vie. An interview
Plural. Six different people living separate lives but each earning their living as associates in the same firm.
Lorsque nous travaillons, il est vrai que nous gagnons bien nos vies. A printed pamphlet.
A group of people each one of them earning a separate living.
Nous travaillons tous les 2 et nous gagnons bien notre vie.
A forum.
Singular. A couple with two separate incomes. But plural would be just as acceptable.
Nous gagnons bien nos vies et nous n'aurions aucun problème à lui venir en aide. A forum.
A couple, the plural implies they both earn a living but it can't be for certain.
.. nous gagnons bien notre vie Article printed in a magazine.
A couple working together, a common source of income. I would not expect the plural here.
Ce genre de bien coûte une petite fortune, et Hugo se réjouit déjà du pourcentage qu'il va empocher. Nous gagnons bien nos vies, en particulier Hugo. (excerpt from a novel)
Plural, and no choice here because of the precisions given on the partner's income.
Pour ce qui est du paiement, comme l'a dit Éloïse, nous gagnons bien nos vies, et nous souhaitons assumer nous-mêmes les frais de notre mariage. Excerpt from a novel
Plural but singular possible here.
Here is an answer to the first question only; the second one has been treated already in several preceding questions (see the linked questions).
In fact "gagnons bien nos vies" is not used at all in print: ngram. It does not mean that nobody will ever use it in their speech; occasionally someone might say that because the context implies that several livings are concerned. We have a similar situation in English; mostly, English speaking people choose to say "we earn our living", but a few say "our livings": ngram. It just happens that the French feel somewhat more strongly that "vie" should be in the singular, or possibly that they have acquired a stronger habit somehow. It's nothing more than an idiomatic particularity. I must say that to my ear "nous gagnons bien nos vies" does not sound usual; however it doesn't sound illogical.
Here "vie" has to be taken as "means of livelihood" (living); nevertheless there is no implication of collaboration: what is meant is "chacun d'entre nous gagne bien sa vie".
On Question 1, this reflects the fact in situations where English has what is called a "distributive plural", French has a tendency to use the singular.
For example,
corresponds to
In this case, each person has one hat. Considering the totality of hats would suggest the plural, whereas considering what each person has suggests the singular.
Even in English, there is a degree of variation, and one of the cases where you have a greater likelihood of finding a distributive singular is in idioms. In English, you'd never say "They can't put their fingers on what's wrong." The idiom is "to put one's finger on something," and this outweighs the fact that "they" collectively have many fingers. Of course, gagner sa vie is an idiom, which makes it even less likely to appear with a distributive plural in French.
With all of this being said, you do find the distributive plural in French. You can say
