Síðan ég fékk stærra geymsluhólf er svefnherbergið mitt miklu minna fullt.

Here síðan means since, introducing a time clause: since I got...

That is different from the very common adverb use of síðan meaning then or after that.

You can tell from the structure:

• Síðan ég fékk... = Since I got...
• Ég fór heim og síðan sofnaði ég. = I went home and then I fell asleep.

So in this sentence, síðan is a conjunction, not an adverb.

Because the sentence talks about:

• a completed event in the past: ég fékk stærra geymsluhólf = I got a bigger storage compartment/unit
• a state that is true now: svefnherbergið mitt er... = my bedroom is...

So the logic is:

Since that past event happened, this present situation is now true.

This is very natural in Icelandic. English often does something similar:

• Since I moved, I’m happier.
• Since I got more storage, my room is less full.

This is because of the Icelandic V2 word order rule.

In main clauses, the finite verb normally comes in the second position. If you begin the sentence with something other than the subject, that whole element takes the first slot.

Here, the first slot is the entire subordinate clause:

Síðan ég fékk stærra geymsluhólf | er | svefnherbergið mitt ...

So even though svefnherbergið mitt is the subject, the verb er comes before it because the sentence starts with another element.

Geymsluhólf is the direct object of fékk.

The verb fá normally takes the accusative, so geymsluhólf is accusative singular here. Since geymsluhólf is a neuter noun, the comparative adjective has to match it:

• stærra geymsluhólf = bigger storage compartment/unit

Why stærra and not stærri?

Because stærra is the neuter singular form.
Stærri is the masculine/feminine singular form.

So:

• stærri bíll = a bigger car
• stærri taska = a bigger bag
• stærra herbergi = a bigger room

It is a compound noun:

• geymsla = storage
• hólf = compartment, section, cubby, slot

So geymsluhólf literally means something like storage compartment.

Depending on context, English could render it as:

• storage unit
• storage compartment
• locker
• storage cubby

Compound nouns like this are extremely common in Icelandic.

Because noun + definite article + possessive is a very common and neutral Icelandic pattern:

• bíllinn minn = my car
• húsið mitt = my house
• svefnherbergið mitt = my bedroom

This is usually the most natural way to say my specific X.

The version mitt svefnherbergi is possible, but it is more marked and often feels more emphatic or contrastive, something like:

my bedroom (as opposed to another room / someone else’s bedroom)

So in this sentence, svefnherbergið mitt is the normal, everyday choice.

Because svefnherbergið is neuter singular.

The possessive minn changes for gender and number:

• masculine: minn
• feminine: mín
• neuter: mitt

So:

• bíllinn minn = my car
• taskan mín = my bag
• svefnherbergið mitt = my bedroom

The possessive agrees with the noun it belongs to.

This is a very useful Icelandic comparative pattern.

• minna = less
• miklu = much / far
• fullt = full

So:

minna fullt = less full
miklu minna fullt = much less full / far less full

A key point: miklu is very commonly used with comparatives as an intensifier.

Compare:

• miklu betri = much better
• miklu stærra = much bigger
• miklu minna = much less

So miklu minna fullt is completely natural grammar.

Because fullt is the neuter singular form of the adjective fullur.

The subject is svefnherbergið, which is neuter singular, and predicative adjectives agree with the subject:

• Herbergið er fullt. = The room is full.
• Bíllinn er fullur. = The car is full.
• Taskan er full. = The bag is full.

So in this sentence, fullt matches svefnherbergið.

It can easily carry that more natural English sense.

Literally, fullt means full, but when talking about a room, it often implies:

• full of stuff
• crowded with things
• cluttered

So while less full is the direct meaning, in natural English you might also say:

• my bedroom is much less cluttered
• my bedroom is much less packed
• my bedroom has a lot less stuff in it

The exact best translation depends on the context.

Yes, you could say:

Síðan ég fékk stærra geymsluhólf hefur svefnherbergið mitt verið miklu minna fullt.

That would mean something like my bedroom has been much less full since I got a bigger storage compartment/unit.

The difference is mainly one of aspect and emphasis:

• er presents the current state very directly
• hefur verið emphasizes that the state has continued over time up to now

In everyday Icelandic, the version with er is very natural here.