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.

Hm. This is somewhat difficult to answer. They essentially mean the same thing but are used at different times depending on the rest of the sentence. Here's a simple distinguishment which I hope will help: Electric refers to anything that runs on electricity i.e. An electric kettle Electrical refers to something related to electricity i.e. electrical faults, electrical component. "Electric" is used in front of a device or machine that runs on electricity. It is used when the object has been specified. "Electrical" is used in a more general sense, as in referring to ambiguous nouns. (By ambiguous, I mean unspecified e.g. machine, appliance). "electrical" is also used when the object is specified, but the object is not run on electricity, but is related to electricity i.e. Electrical engineer

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.