Do "How come?", "How do you mean?" and "Why's that?" mean the same? Do "What sort/kind of ... ?" and "Such as?" mean the same?

“How come?” with “How do you mean?” and vice versa.

Both interrogative sentences have "how", which means in what manner(=how) and even "how" only can also communicate (almost) the same meaning. It depends on the context, situation, and the language users' habit, manner(s)(=courtesy), or discretion.

Q1. I’m going to my brother’s wedding, but I’m rather nervous about the whole thing, though.

Really? How come? VS. How do you mean? (You could use both questions, BUT 'How do you mean?' has more the sense of 'In what way are you nervous?')

Q2. Have you got any holiday plans? No, no holidays this year, I’m afraid.

Why’s that? VS. How come? or How do you mean? (How come is okay but How do you mean? is a little different because the answerer already said very clearly that he had no plan this year)

Q3. I’ve put in an offer on a flat and I’m saving up. Wow! What’s it like? Well, it needs a lot of work doing to it. Such as? Oh, everything. …… By the way, I’m starting my own training business. What sort of training? Presentations, mainly.

Is it OK to replace “Such as ?” with “What sort of work?” and “What sort of training?” with “Such as?” (Yes, it's okay; such as means for example? what? what sort of things? what kind of thing? Like what?)

Q4. After the wedding, I’m going on a trip. Are you? What sort of trip? …. I’m learning new languages at the moment. Are you? What kind of languages?

Is it OK to replace “What sort/kind of trip/languages?” with “Such as?”

After the wedding, you go on a honeymoon trip. So asking what sort of trip is not natural and if someone is learning a language, you can just ask "what language are you learning?, or just what language?"

"such as" is used when you mention something and want to give some examples.

I have books such as comic books, novels, science-fictions, etc. So if it's not necessary to enumerate a list of things, it's better to ask directly or exactly but politely what you want to ask about.