Learn English – “ON the west of New York?” Is this preposition correct

It would be ridiculous to suppose there's some kind of "critical distance" involved here, whereby people nearer would always speak of going into the city, and people further away would simply go to the city.

It's just that into more strongly implies going right inside the place (usually, the city or town centre, in this case). It's a metaphor of location, invariably implying a central part of the stated destination.

Going to a place is a metaphor of direction (to = towards, in the direction of). If you're going to London, your intended destination is indeed some part of London, but it might not be a particularly central part.

I live less than 1/2 a mile from my local town centre, but it's a fairly large town, and even in the opposite direction people well over a mile away are still considered to live in this town. And we all speak of going into town when we mean going to the town centre.

Los Angeles is famously "spread out" (the area is almost as big as London, but the population is less than half). People living in outer parts of London commonly speak of going into London to mean going to a much more central part, and I expect Angelenos would be at least as likely to adopt such usages.

So the short answer is you go to a place that's somewhere other than where you started (the preposition represents the direction you'll be going in). You go into a place when your destination is somewhere that could reasonably be described as "inside" that place (the preposition represents your spatial location).

It depends. If you mean a one week period of time, and you are still there, then:

I have been in New York with my parents and my sisters for one week.

If you mean the 7 days prior to the statement, then:

I have been in New York with my parents and my sisters since one week ago.

I prefer have been since you are referring to a period of time.