This is basically correct. Without the other context, the first sentence sounds a bit strange. Who is she? The only thing you would add is "I'd like to stay here and live with my family FOR the rest of my life." which is more proper but your sentence is not incorrect and any English speaker would know what you mean.

How do you usually express the "that'd" when speaking to another? It is unspeakable!

In formal writing, like for an article or a business letter, it is correct practice to write out contractions: - I would like to accept your company's employment offer. - If tomorrow is not convenient, please specify a time when you would be available. In speech and in informal writing, contractions are very common and often preferred: - I'd like to take you up on your offer of twenty bucks for my old lawnmower. - If you're not free tomorrow, let me know a time that'd work for you.

The use of I'd is very common in native English speakers.

The word I'd is a contraction of the words I and would, so it means the same thing. Maybe if you were writing a formal essay or something for school you would use "I would" but in everyday speech "I'd" is just fine.