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.)
Best Answer
The sentence is incorrect, e.g. "impression between students".
So I would say
(in case you are pursuing it now, and not in the future)
You don't need a comma, except possible after "back home". Commas represent pauses in conversation, so in order to make the text more eligible, you add commas.
For the second question the answer I would say is