1. I think Mei wanting to get a man has many sides.
- While she is the Mizukage, and is respected for her position, the villagers also want her to be happy.
- She possesses two bloodline limits, which are valuable in the ninja world, and passing those on to a son or daughter would be important for the next generation.
- I think that there is some pressure with her strength and position, which makes it hard for her to have that sort of romantic, picture book wedding that she dreams about.
2. I think Mei and Kakashi did meet, they were both Kage at the same time. But we don't know anything about what happened during those years when Kakashi was the Hokage, so we don't know what their interactions were like.
3. It's kind of a dumb joke; however, it is a thing where women who are ~30 years old and not in a stable relationship do have certain anxieties about themselves and their place in society. Which is a weird thing to give a character in a manga written for teenage boys.