Samband okkar helst sterkt, þó að við búum í mismunandi löndum.

Okkar is the genitive form of við (we) and it is used to mean our.

So samband okkar literally means relationship of us, i.e. our relationship.

In Icelandic, possessive words like minn, þinn, okkar, ykkar can often come after the noun. That is very natural here.
Okkar samband is also possible, but samband okkar is a very normal way to say it.

Not necessarily. Samband can mean relationship, connection, association, contact, depending on context.

In this sentence, it most naturally means relationship or bond. That could be romantic, but it could also describe a close personal connection more generally.

Here helst is a verb, not an adverb.

It comes from haldast, which means to remain, stay, keep holding, last in this kind of context.

So:

• Samband okkar helst sterkt = Our relationship remains strong

A learner may also have seen helst meaning preferably or especially, but that is a different use. In this sentence, it is clearly the verb because it goes with the subject samband okkar.

Because samband is a neuter singular noun.

The adjective sterkur (strong) has to agree with the noun it describes. Here it is used as a predicate adjective after the verb, but it still agrees with the subject:

• masculine: sterkur
• feminine: sterk
• neuter: sterkt

So:

• samband = neuter
• therefore sterkt

Þó að means although, even though, or though.

It introduces a contrast:

• Samband okkar helst sterkt = one fact
• þó að við búum í mismunandi löndum = a contrasting fact

So the whole sentence means something like:

• Our relationship remains strong, even though we live in different countries.

Yes, very often you can.

Þótt and þó að are commonly used with a very similar meaning: although / even though.

So a version like:

• Samband okkar helst sterkt, þótt við búum í mismunandi löndum.

would also be natural.

Búum is the 1st person plural present form of að búa, meaning to live / reside.

The subject is við = we, so the verb must match:

• ég bý = I live
• þú býrð = you live
• hann/hún/það býr = he/she/it lives
• við búum = we live

So:

• við búum = we live

Many learners are told that þó að often goes with the subjunctive, and that is a useful thing to know.

In this sentence, though, búum looks the same in the present indicative and present subjunctive for við. So the form itself does not show a visible difference.

That means you may see this explained as a subjunctive environment, but the actual word form búum would be the same either way.

Because í takes the dative when it means in a place, with no movement involved.

Here the meaning is location:

• we live in different countries

So Icelandic uses the dative plural of land:

• nominative plural: lönd
• dative plural: löndum

That is why the sentence has:

• í mismunandi löndum

If there were movement into something, Icelandic could use the accusative instead, but this sentence is about location, not motion.

Mismunandi often behaves like an indeclinable descriptive word in modern Icelandic, so it usually does not change form for gender, number, or case.

That is why you get:

• mismunandi land
• mismunandi lönd
• mismunandi löndum

The noun changes, but mismunandi stays mismunandi.

Because the meaning is indefinite and general:

• in different countries

not

• in the different countries

Icelandic often leaves out the definite article in cases like this, just as English does.

Yes. In Icelandic, a subordinate clause introduced by something like þó að is commonly separated by a comma.

So this punctuation is completely normal:

• Samband okkar helst sterkt, þó að við búum í mismunandi löndum.

A few parts often stand out:

• þ in þó is like the th in thin
• ð does not appear in this sentence, but learners often compare it with English th in this
• ó is a long vowel, roughly like o in English go, but not exactly the same
• búum has two syllables: bú-um
• löndum has the rounded vowel ö, which English does not really have

A rough learner-friendly approximation might be:

• Samband okkar helst sterkt, þó að við búum í mismunandi löndum
• SAM-band OK-kar helst STERKT, tho ah vith BOO-um ee MIS-mu-nan-di LERN-dum

That is only an approximation, but it can help you get started.