"I'm having lunch with friends next week."
That is fine too.
Just a little note, as there have been quite a few questions about the position of time. When we place the time factor in the very front of a sentence, it applies to the whole succeeding clause 我和朋友吃午饭, but when we place it before the verb, it applies just to the verb. In the latter case 我和朋友 and 吃午饭 become more distant and the correlation is slightly weakened. 我和朋友 means "I and my friend(s)" and the notion of “eating with” or "together" is only implied. An extreme (though unreasonable) interpretation will be both me and my friend(s) will be having lunch, vs not, but separately, during next week. The listener does need more speculation to understand the sentence.
I think I understand what you're saying here but I'm not sure what your example means. Are you saying it would be wrong to move the adverbial phrase to the end in that case because it would result in a repeated 一起 at the end of that sentence or is there a general rule regardless? In English, "We put toys together together" is grammatical and rather easily understandable.
Grammatically there is no problem. Yet we need to take note placing 会 before 吃午饭 would make the sentence conceived as [Next week] > [I and my friend] > [will] > [have lunch]. It would be weird as everybody would normally have lunch on every day in any week and it makes no sense to talk about it. So we would say instead 下个星期我会和朋友吃午饭, and the breakdown would be [Next week] > [I] > [will] > [have lunch with my friend], Having lunch with my friend is a specific event to me and it makes sense that I say it will happen.
Compare with 下个星期我和朋友会去La fourchette吃午饭 / My friend and I will have lunch at La fourchette next week. The breakdown would be [Next week] > [I and my friend] > [will] > [have lunch at la fourchette]. This would also make sense in Chinese.
Building a Chinese sentence is like decorating a Christmas tree. You can put an ornament on one branch rather than another. There are many ways to do it but some ways looks odd and some others are beautiful. The basis here is 我吃午饭 and the other words are ornaments.
No it is not strange. At times it would be optional or required that you use 我的 before 朋友 to make an emphasis that it is about "my" friends rather than "your" friends or some other people's friends. It is all about reducing the chance of misunderstanding each other in a given context.