With go, or verbs that can be interchanged with go, such as drive, walk, move, etc., sometimes the relation between the destination and speaker can be expressed with either verb.
I went down to the store.
I went up to the store.
Do we consider the store "down" or "up" from us? Assuming the store is not significantly downhill/uphill from you, it doesn't matter too much. In my opinion, "down" may have a bit of meaning of "forward" or "in front of," - something I've commonly heard is "I'm going to walk down this path" when you really mean "I'm walking forward along the path."
Conversely "up" will have a bit of meaning of "behind." This isn't really a hard and fast rule and could vary widely depending on region.
The implication in using either word is that a bit of a journey was involved or will be involved.
Native speakers would understand either way, unless the "X" in "I went down/up to X" is physically higher or lower than you. Saying "I'm going down to the third floor" when you are on the second floor will confuse people. Saying "I'm going down to the restuarant" will not.
In A4, assuming a situation of theater seats where they are all in a row (none are really "up" or "down") - you could substitute "up" and not really change the meaning, though "down" sounds more natural.
A4. Would you mind moving (further) up so that we can sit here, too?
Regarding 5a, "up" or "up to" can also mean "near," particularly with words that involve the subject moving items (pull up, push up), or the subject moving himself/herself/itself ("up to" will be used - ran up to, walked up to, etc.)
First, you have to write "the sea". Also, this is not an issue of grammar but meaning.
More context is necessary to choose, but in general:
He went to the sea alone.
This means you went to the area near the sea. If I went to a coastal area, I could say this. But it does not really mean in/on the water.
He went into the sea alone.
This means he went from being "out of the water" to "in the water".
He went in the sea alone.
Same as "into" when using went, since motion is indicated.
Generally I would use on the sea when you are traveling in a boat.
Best Answer
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).