Would plain 'zenme' without zenmeyang also work?
Zenme makes more sense than zenmeyang here.
I feel that 样 makes it sound a little weird... When i speak, I would leave it out normally...
Yes, it works!
Can I not put 怎么样 at the end?
That would mean "how about we improve..."
I don't think so.
Isn't the order 怎么样我们提高汉语水平 also correct?
No. It would be logical to put a comma after 怎么样, making 怎么样，我们提高汉语水平 which means "how about this: we improve our Chinese".
怎么样我们提高汉语水平？ Is it a big mistake 怎么样 on the first place?
The suggested answer used words that aren't even an option.
Can we say zenmeyang women... Instead of women zenmeyang... ?