La ambasado estas proksime; aliflanke, la konsulejo estas en alia urbo.

Because proksime is an adverb, meaning nearby or close by.

In this sentence, La ambasado estas proksime means The embassy is nearby. English also uses an adverb here: nearby.

If you used proksima, that would be an adjective, and it would usually describe a noun more directly, for example:

• La ambasado estas proksima al la stacidomo. = The embassy is close to the station.

So:

• proksime = nearby, in a nearby place
• proksima = close, near, describing a noun

Both can be correct in the right context, but here proksime is the natural choice.

It can do both.

• By itself: La ambasado estas proksime. = The embassy is nearby.
• With de: La ambasado estas proksime de la hotelo. = The embassy is near the hotel.

So if you just want to say something is nearby in general, proksime is enough.
If you want to say what it is near, use proksime de ....

Aliflanke means on the other hand, by contrast, or conversely, depending on context.

In this sentence, it introduces a contrast:

• La ambasado estas proksime; aliflanke, la konsulejo estas en alia urbo.

So the idea is:

• the embassy is nearby,
• but in contrast, the consulate is in another city.

It is a common linking word for comparing two different situations.

Esperanto often combines elements into a single word when they form one clear idea.

Aliflanke is built from:

• ali- = other
• flank- = side
• -e = adverb ending

So literally it is something like on another side, but idiomatically it means on the other hand.

Learners often want to separate it, but writing it as one word is normal Esperanto usage.

Yes, in many situations you could use sed, but the nuance is slightly different.

• sed = but
• aliflanke = on the other hand / by contrast

So:

• La ambasado estas proksime, sed la konsulejo estas en alia urbo.

is perfectly understandable and natural.

However, aliflanke sounds a bit more like you are setting up a comparison between two sides of a situation. It is a little more formal or structured than simple sed.

The semicolon separates two closely related independent clauses.

The two parts are:

• La ambasado estas proksime
• aliflanke, la konsulejo estas en alia urbo

Each part could stand as a sentence by itself, but they are closely connected in meaning, so the semicolon works well.

In Esperanto, just as in English, punctuation is fairly flexible here. You could also see:

• La ambasado estas proksime. Aliflanke, la konsulejo estas en alia urbo.
• La ambasado estas proksime, sed la konsulejo estas en alia urbo.

So the semicolon is not a special grammar rule; it is mainly a style choice.

Because both are being treated as specific, identifiable things: the embassy and the consulate.

Esperanto la works much like English the.

So:

• la ambasado = the embassy
• la konsulejo = the consulate

The speaker is not talking about any embassy or any consulate, but specific ones known from context.

Because alia urbo means another city, not the other city.

Compare:

• alia urbo = another city
• la alia urbo = the other city

In the sentence, the idea is just that the consulate is in some different city, so alia urbo is the natural choice.

Konsulejo means consulate.

It is formed from:

• konsulo = consul
• -ej- = place associated with something or someone
• -o = noun ending

So konsulejo is literally a place connected with a consul.

This is a very common Esperanto pattern. For example:

• lerni = to learn
• lernejo = school, a place for learning

So learning -ej- is very useful.

Yes. En means in, and it is the normal preposition for being located inside a city, country, room, building, and so on.

So:

• en urbo = in a city
• en Londono = in London
• en alia urbo = in another city

Here, La konsulejo estas en alia urbo simply means that the consulate is located in another city.

Yes. Esperanto word order is fairly flexible, especially with adverbs and linking expressions.

For example, you could also say:

• La ambasado estas proksime; la konsulejo, aliflanke, estas en alia urbo.

This gives slightly different emphasis, but the meaning is basically the same.

The version in your sentence puts aliflanke right at the start of the second clause, which is a very natural place for a contrast marker.