For first question, I say “That’s wrong.”
Men also use 私 in formal situation. For example business scene or you have to speech in front of audience, some thing like that. Japanese president Abe, he always use 私. Even business man use 私 in a conference or talk with client. 俺 is casual pronoun. So you can use daily but usually don’t use it in important business scene or any formal situation.
Men has 3 mejor pronouns. 俺＜僕＜私。俺 is informal. 私 is formal. 僕 is between. You can use daily also you can use formal situations. But it more casual sound than 私, so it’s depends on how formal is that situation. Actually if you read Japanese manga, you can see more pronouns lol But most Japanese men use 俺/僕/私, so these are enough.
Then, about group. It’s depends on the speaker is man or woman, also it’s formal situation or not.
If the speaker is woman, it’s always "私たち" even only the speaker is woman in the group. And it doesn’t matter formal or informal. Because women always use 私.
If the speaker is man, the speaker use 俺たち for casual situation. 私たち for formal. It doesn’t matter even the speaker is only man in the group.
We don’t care about that group is only men or women or mix. The point is what is the speaker’s sexuality.
I hope it helps your understanding.