Í dag finn ég nefnifallið strax, en í gær ruglaði ég því saman við þolfallið.

This is because Icelandic normally follows the V2 rule in main clauses: the finite verb comes in the second position.

• Í dag is the first element.
• So the verb finn must come second.
• The subject ég comes after the verb.

So:

• Í dag finn ég ...
• Í gær ruglaði ég ...

If you start with the subject instead, you get normal subject-first order:

• Ég finn nefnifallið strax í dag.

Both are correct, but the original sentence puts emphasis on today and yesterday.

Yes, that is the easiest way to treat them.

• í dag = today
• í gær = yesterday

Even though í often means in, these are very common fixed time expressions, so it is best not to translate them too literally every time.

Yes, it comes from finna, which often means to find, but in this sentence it is being used in a broader sense: to identify, recognize, or spot.

So finn ég nefnifallið strax means something like:

• I spot the nominative right away
• I recognize the nominative immediately

It is not necessarily about physically finding something.

That -ið is the suffixed definite article in Icelandic. Icelandic often attaches the to the end of the noun instead of using a separate word.

So:

• nefnifall = nominative
• nefnifallið = the nominative

and

• þolfall = accusative
• þolfallið = the accusative

These words are neuter singular, and -ið is the normal definite ending here.

Because the word nefnifall is just a noun that names a grammatical case. Its meaning is nominative, but its role inside this sentence is still determined by normal Icelandic syntax.

Here, nefnifallið is the direct object of finn, so grammatically it is in the accusative.

The reason it still looks like nefnifallið is that for many neuter nouns, the nominative and accusative singular have the same form. So the form does not change visibly.

In other words:

• the word means the nominative
• but in this sentence it functions as an object

Those are two different things.

Because the expression rugla einhverju saman við eitthvað uses the first thing in the dative.

The pronoun það changes like this:

• nominative/accusative: það
• dative: því

So in:

• ruglaði ég því saman við þolfallið

the því is in the dative because of the verb pattern.

This is something you usually just have to learn together with the verb:

• rugla einhverju saman við eitthvað = to confuse one thing with another

It is a common Icelandic expression meaning:

• to confuse X with Y
• to mix up X with Y

So:

• rugla því saman við þolfallið = confuse it with the accusative

The word saman is part of the idiom here. Literally it suggests together, but in real usage you should learn the whole phrase as one unit:

• rugla einhverju saman við eitthvað

Because við takes the accusative in this expression.

So the structure is:

• rugla einhverju saman við eitthvað

That means:

• first thing: dative
• second thing after við: accusative

So við þolfallið is exactly what you would expect here.

Because Icelandic adverbs like strax can move around somewhat, and this placement is natural.

Here:

• finn ég nefnifallið strax

means I recognize the nominative right away.

You could also hear or say:

• Í dag finn ég strax nefnifallið

That is also possible, but the original version sounds very natural and keeps strax closely connected to the whole action finn ....

Because the sentence has two coordinated clauses joined by en:

• Í dag finn ég nefnifallið strax
• en í gær ruglaði ég því saman við þolfallið

Each clause has its own verb and its own subject, so Icelandic normally states ég again.

This is very similar to English:

• Today I recognize the nominative right away, but yesterday I confused it with the accusative.

You would normally repeat I in English too.