What's "network" mean here? Is it like a tv channel?

What’s the meaning of this whole sentence? She can call him by any name which she wants?

Yes, this sentence is him being flirtatious by telling a woman that she can call him any name (meaning a nickname) when they meet up in person.

Why does she say "get" not "give"? Is there any difference in nuance?

I can understand how these two words can be confusing! When used as a verb, "get" means to receive something from someone else. "Give" means to freely supply something to another person, such as a gift or item. Grammatically, the sentence is "Will you(subject) get (verb) us (indirect object) better gifts (direct object)?" Phoebe uses "get" instead of "give" because she wants a better present from Chandler. She doesn't want to receive the crappy present that he got for her. Even though Chandler (the subject) is the one who is giving the presents, Phoebe wants to receive a present she will actually enjoy, which is why she says "get" instead of "give." Ex: Could you get me a paper towel please? Ex: He gave me some advice about applying for jobs. Ex: I need to get a new phone. Ex: She was given a new computer for her birthday.

Can I say "I'm very so sorry."?

No, this is an exaggerated form of an apology and the "so" must go before the "very." The only way to use this expression is to say "I'm so very sorry." In other cases where you wish to use "so" and "very" together, the "so" goes first as well. Ex: I'm so very sorry for accidentally hitting your car. Ex: I'm so very sorry for coming late.

What's "network" mean here? Is it like a tv channel?

Kind of! A TV "network" is a company or group that distributes shows and program content to one or many television stations and providers. So they distribute the content that gets shown on TV channels! Ex: Another network signed a contract with us to distribute our show! Ex: You can only watch this show with TV providers who are a part of OBH Max network.

Is this expression "funny how things worked out" used a lot as this meaning?

Yes, "funny how things worked out" is an idiomatic expression that is used to mean it is interesting, comical, or ironic how something ended up to be. It is always used for this meaning. You can also say "funny how thing turned out". Ex: The bullies in school are now working for me. Funny how things worked out, isn't it?