Learn Hardcore Russian: По карте видно, что наш дом на востоке города, а университет на западе. - From the map it is clear that our house is in the east of the city and the university is in the west.

по
- dative (карта → карте) often means:
  - “according to” (по расписанию – according to the timetable),
  - “by means of” (по телевизору – on TV),
  - “based on” something.

So по карте видно = “it is visible/clear from the map”, i.e., if you look at the map, you can see that…

Why is it видно and not something like мы видим or можно увидеть?

Видно here is an impersonal predicate, not a normal verb with a subject.

видно = “it is visible / it is clear.”
There is no grammatical subject (no “it,” no “we”).
Russian often uses impersonal structures where English uses “it is …” or “you can …”.

Compare:

По карте видно, что…
“From the map it is clear that…”
Мы видим по карте, что…
“We see from the map that…” (more explicit “we”)
По карте можно увидеть, что…
“From the map one can see that…” (more formal, slightly heavier)

The original is the most natural and neutral: По карте видно, что….

Is there a missing subject here, like “it” in English (“It is visible from the map”)?

In English, we’d say “It is visible from the map that…”, with dummy it.

In Russian, impersonal constructions do not need any subject at all.
Видно already functions as a complete predicate:

По карте видно, что…
literally: “From the map [it] is visible that…”, but there is no it in Russian grammar.

So nothing is “missing”; the structure is just impersonal by design.

Why is there a comma + что in видно, что наш дом…? Is что a conjunction here?

Yes. Here что is a subordinating conjunction meaning “that”, not the question word “what.”

По карте видно, что наш дом…
“From the map it’s clear that our house…”

Comma rule:
In Russian, a subordinate clause introduced by что is normally separated by a comma from the main clause:

Я знаю, что он дома. – I know that he is at home.
Видно, что наш дом на востоке города. – It’s clear that our house is in the east of the city.

Why is there no verb like есть or находится in наш дом на востоке города?

Russian often omits “to be” in the present tense in such sentences.

наш дом на востоке города
literally: “our house in the east of the city.”

English must add “is”:
→ “Our house is in the east of the city.”

In Russian you can add a verb:

Наш дом находится на востоке города. – neutral, a bit more formal: “is located.”
Наш дом есть на востоке города. – sounds wrong/unnatural here; есть as a present “to be” is rarely used this way in modern Russian.

So the bare noun + location phrase is the standard present‑tense pattern.

What case is на востоке and на западе, and why are they in that case?

Both востоке and западе are in the prepositional case:

восток → на востоке
запад → на западе

With на + prepositional, you typically express location (where?), not direction:

на востоке – in the east (of something)
на западе – in the west (of something)

So:

наш дом на востоке города = our house is located in the east of the city.
университет на западе = the university is in the west (of the city, from context).

Why is it города in на востоке города, not город?

Города is genitive singular of город.

The pattern “на востоке чего?” uses the genitive to say “in the east of X”:

на востоке города – in the east of the city
на севере страны – in the north of the country
на юге России – in the south of Russia

So the structure is:

на востоке (чего?) города – “in the east of the city.”
The thing whose east it is goes into genitive.

Why doesn’t университет на западе repeat города? Is something omitted?

Yes, города is simply understood from context and omitted to avoid repetition.

Fully explicit:

…наш дом на востоке города, а университет на западе города.

Since города already appeared, it’s natural to drop it the second time:

…наш дом на востоке города, а университет на западе.

Listeners automatically understand “на западе (города)” – in the west of the city.

What is the difference between а and и here? Could I say …городa, и университет на западе?

You could say и, but а is better here.

и = “and” in the sense of simple addition, no contrast.
а = “and/but” in the sense of contrast / comparison between two things.

In this sentence, we are contrasting locations:

наш дом на востоке города, а университет на западе.
“Our house is in the east of the city, whereas the university is in the west.”

Using и would sound more like just listing facts, without emphasizing the east–west contrast:

…городa, и университет на западе. – grammatical, but weaker contrast.

Can we change the word order, for example На востоке города наш дом? Does it sound different?

Yes, you can change it, and it slightly shifts the emphasis.

Наш дом на востоке города – neutral; subject first.
Focus is simply on the fact where the house is.
На востоке города наш дом – emphasizes “In the east of the city” (that’s where our house is).
This can sound more contrastive or descriptive, e.g., if you are listing what is where.

In your full sentence, all of these are grammatically fine:

По карте видно, что наш дом на востоке города, а университет на западе. – neutral.
По карте видно, что на востоке города наш дом, а университет на западе. – puts stronger focus on “in the east of the city.”

In everyday speech, the original word order is the most typical.

What’s the difference between по карте видно, на карте видно, and из карты видно?

All can appear, but they have slightly different flavors:

По карте видно, что…
- Most natural and idiomatic here.
- Means “From the map / On the basis of the map it is clear that…”
- Very common in speech.
На карте видно, что…
- Literally: “On the map it is visible that…”
- Focuses more on what is actually drawn/shown on the map.
- Often used when you’re talking about something literally visible on the map’s surface.
Из карты видно, что…
- More bookish/formal.
- Sounds like academic/technical writing: “From the map it follows that / it is evident that…”
- Less common in everyday speech.

In a normal conversation, По карте видно, что… is the most natural choice.

How do I say “to the east of the city” (direction) instead of “in the east of the city” (location)?

Important distinction:

на востоке города – static location: in the eastern part of the city.
на восток от города – direction: to the east of the city (outside or beyond it).

Examples:

Наш дом на востоке города.
Our house is in the east of the city.
Деревня находится на восток от города.
The village is to the east of the city (separate from it).

So if you want direction / position relative to the city, use:

на восток от города / на запад от города.

What is the register and tone of this sentence? Is it formal, informal, or neutral?

The sentence is neutral and standard. You can use it:

in everyday conversation,
in textbooks,
in written descriptions (e.g., explaining a map).

Words like по карте видно, наш дом, университет, на востоке города, на западе are all common and stylistically neutral.