It should be:

Mon ami et moi mangeons.

Mon ami et moi, nous mangeons. (note the comma)

The personal pronoun nous can be omitted here. I don’t know why but the sentence sounds peculiar to me. I guess it could be part of a bigger sentence.

Colloquially:

Mon ami et moi, on mange

On cannot be omitted.

As another example:

Mon ami et moi avons une grande maison.

Mon ami et moi, nous avons (on a) une grande maison.

Yet another:

Mon ami et moi sommes tombés amoureux. Ça peut marcher?

(Vu ici: http://www.internationalnewsblog.com/2019/02/mon-ami-et-moi-nous-sommes-tombes-amoureux.ca-peut-marcher.html)

I
“Mon ami et moi mangeons” and “Mon ami et moi, nous mangeons” are not idiomatic unless you want to say that you are beings that have to take in food in order to live. Idiomatically, if you want to say that you and your friend are having a meal or that you are eating a snack, you’d have to say something like this;

• Mon ami et moi sommes en train de manger. or using “on”
• Je suis avec mon ami, on mange.

You could say this but it is not very idiomatic; it sounds strange;

• Je suis avec mon ami, nous mangeons maintenant.

Fully idiomatic is this;

• Je suis avec mon ami, nous sommes en train de manger.

II

“Mon ami et moi mange” is not correct grammatically.

“Moi et mon ami” is a possibility nowadays; there used to be a rule of politeness saying that the person speaking should be mentioned last (mon ami et moi), but people seem not to bother much about it any more.