"hey, how are you" in he middle of an email after a request is very strange. Asking for something and finishing with the non committal "let's get together soon" is also a red flag that you might want something from me from a business perspective but are only feigning interesting from a personal perspective.

That reads really weird. Putting a greeting in the middle of an email is jarring.

I disagree with this but it may just be a style thing. Reading "Hey, how are you?" in the middle of the content put me on guard, I feel like I might be being manipulated somehow.

This sounds very transactional, the personal concern seems to be an aftertoghth.

I was struggling with this dilemma recently, when wrting an email. Knowing what you just said, that your request should be in the first paragraph, and at the same time wanting to show humman connection, I don't think that I did very good job. Still haven't found a smooth way to express both things in the first sentance/paragraph.

I guess, it depends on the person you are writing to.

I really rail against people who think 'This.' is a sentence, let alone a useful contribution.

It's especially galling when the topic is 'Put your question / point in the first sentence / paragraph' and someone's first word, sentence, and paragraph is simply 'This.'