I need to write a dialogue using an informal registertone.

"register" kind-of has the same meaning as "tone", but it's really only in more technical jargon (especially in computer science). "Tone" is used for writing and speaking.

Why, it's you!

"Why, it's you!" Is a common English phrase, and their is always a little pause after "why" (at least when spoken).

Hey Katie, meet Ann.

Same here, it's natural to pause after saying "hey <someone>".

Remember when/what I told you about my cousin?

I am so glad to finally meet you!

Another common phrase is "It's so good to finally meet you!"

EI'm excited to meet you!)

Or "I've been excided to meet you!"

Quite well, thanks.

It's pretty common (and generally more polite) to say "... and you?" after someone asks "how you're doing". e.g.
"how are you doing?" -> "good, and you?"

WSure, why not?

You can't really use "why not?" without first saying "sure" (in this context). Otherwise it comes of as slightly rude as it sounds like you're trying to say "obviously. Why wouldn't you be able to call me that?"

Ann: I need to go now, let's all meet later./again!

Katie: Sure/Okay.

I think you are awesome!

Haha, this is funny, but I don't think most people would say this in English (unless they were trying to be funny and saying it in a really exaggerated manner, or you're screaming it out to a celebrity). It sounds too direct. You can lessen this directedness by first saying "by the way," and maybe instead of "awesome" you use "cool". e.g.
"oh, by the way. I think you're cool/pretty cool." (Without the exclamation mark (!))

Ann: I think so (I think the same wayyou too (I feel the same about you too) 😉.

"I think so too" sounds like Ann is agreeing with Katie and also thinks that "Ann is cool" (which will come of as selfish). You could make that line of thinking work if you say it in an overly sarcastic manner, but it would probably be better to say something like "I know I am" instead of "I think so too".

How is it going?

This is OK, but since you want an informal register, I would definitely use the contraction "How's it going?"

Why not?

This is grammatically fine, but "Sure" and "Go ahead" are much better in this context.

Ann: I think so (I think the same way) 😉.

This sounds like Ann thinks Ann is awesome. I think you want to say Ann thinks Katie is awesome, right? In that case, you could say, "So are you!"