Rotþróin er full.

Because Icelandic loves compound words. The base noun is rotþró and the definite article is added directly to the end, giving rotþróin.

So structurally it is:

• rotþró = septic tank / cesspit
• -in = the (here, the feminine singular definite ending)

That means rotþróin literally works like the septic tank.

In Icelandic, the is usually not a separate word. Instead, it is attached to the noun as a suffix.

So:

• rotþró = a septic tank / septic tank
• rotþróin = the septic tank

This is one of the big differences from English. English uses a separate article; Icelandic very often adds it onto the noun.

Rotþró is a feminine noun.

You can tell in this sentence mainly because of the way the adjective appears:

• masculine: fullur
• feminine: full
• neuter: fullt

Since the sentence says full, the noun it refers to must be feminine singular here.

The definite ending -in is also consistent with a feminine singular nominative form.

Because Icelandic adjectives must agree with the noun they describe in gender, number, and case.

Here, rotþróin is:

• feminine
• singular
• nominative

So the adjective also has to be feminine singular nominative:

• fullur = masculine
• full = feminine
• fullt = neuter

Since rotþróin is feminine, full is the correct form.

Yes. Rotþróin is in the nominative, because it is the subject of the sentence.

And yes, full matches it. After the verb vera (to be), predicate adjectives normally also appear in the nominative.

So in Rotþróin er full:

• rotþróin = nominative subject
• full = nominative predicate adjective agreeing with the subject

A useful contrast:

• Rotþróin er full. = The septic tank is full.
• Ég tæmi rotþróna. = I empty the septic tank.

In the second sentence, rotþróna is no longer the subject, so the form changes.

Er is the 3rd person singular present form of vera (to be).

That is the form used with:

• hann = he
• hún = she
• það = it
• singular nouns like rotþróin

So:

• ég er = I am
• þú ert = you are
• hann/hún/það er = he/she/it is

Since rotþróin is a singular noun, er is the correct verb form.

It would be:

• Rotþró er full.

That is the indefinite version.

In practice, English and Icelandic do not always use definiteness in exactly the same way, so whether you choose rotþró or rotþróin depends on context. But grammatically, the difference is:

• rotþró = a septic tank / septic tank
• rotþróin = the septic tank

A rough guide:

• stress usually falls on the first syllable of each word
• þ is like th in thing
• r is usually tapped or rolled
• ó is a long vowel
• ll in full is often pronounced something like tl to English ears

So a very rough English-friendly approximation is something like:

ROT-throh-in er futl

That is only an approximation, but it points you in the right direction.

Yes, but then you normally add af plus the thing it is full of.

For example:

• Rotþróin er full af vatni. = The septic tank is full of water.

The adjective still agrees with the noun:

• fullur / full / fullt

and then you add:

• af + dative

So in your original sentence, full simply means full on its own, but it can also be expanded into full af ... = full of ...