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1. What is the difference between these two sentences?

I want to meet you.

I want to meet with you.

2. Are these sentences right?

Please let me know if getting better.

Please show me if done right.

3. When you are responding to a question like "How are you doing?", how would you answer?

I am doing well. or I am doing fine. ???

Is there any difference between the words "fine" and "well" in the context of responding to such a question?

Did I correctly punctuate the underlined part?

4. I got this sentence from a post here and why is there an article there?

See, I am not a??? "he."
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3. Either is fine.

4. It was my post, as you know. Several times people have referred to me as "he," so it was a way of saying I am not a man.
Comments  
1. What is the difference between these two sentences?

I want to meet you.

This might be said to someone you haven't met in person before. Or it could be used with more specific information (a particular place and time). For instance:

I want to meet you when you come to (my city). I've enjoyed chatting with you on-line.

I want to meet you at the restaurant at 7:00.

I'll meet you at the restaurant at 7:00.

I want to meet you after work.

I'll meet you after work.

I want to meet with you.

To "meet with" often conveys the idea that someone wants to talk about something in particular. Maybe people are or will be working on something together. Or maybe there's a possibility that they will be arranging something.

For instance:

I want to meet with you about our project.

I want to meet with you to go over some things (check and consider some things).

2. Are these sentences right?

Please let me know if getting better.

You need some kind of an object. Who or what may or may not be getting better?

For instance:

Please let me know if Jim is getting better. (feeling better, recovering from an illness)

Please let me know if your cousin is getting better. (feeling better, recovering from an illness)

Please let me know if your relationship with your boss is getting better. (getting along better with your boss, improvement in the way you and your boss work together)

Please let me know if your Spanish is getting better. (someone is studying Spanish)

Please show me if done right.

You need an object here, too.

For instance:

Please show me if it is done right.

Please show me if this work is done right.

Please show me if my homework is done right.
 BarbaraPA's reply was promoted to an answer.