Learn Hardcore Icelandic: Ég kaupi bláber í búðinni á horninu. - I buy blueberries at the shop on the corner.

Kaupa is the infinitive (dictionary form) meaning to buy.
Kaupi is the present tense, 1st person singular form: I buy / I am buying.

Present tense of kaupa:

ég kaupi – I buy
þú kaupir – you (sing.) buy
hann / hún / það kaupir – he / she / it buys
við kaupum – we buy
þið kaupið – you (pl.) buy
þeir / þær / þau kaupa – they buy

So in the sentence, Ég kaupi… = I buy / I am buying…

Is bláber singular or plural here, and why doesn’t it have a plural ending?

Bláber is a neuter noun whose nominative and accusative singular and plural have the same form: bláber.

So bláber can mean:

one blueberry (in some contexts)
blueberries (more than one)
or even blueberry in a general/mass sense (like English I like blueberry as a flavor)

You normally see the number or context making it clear:

Ég borða eitt bláber. – I eat one blueberry.
Ég borða mörg bláber. – I eat many blueberries.
Ég kaupi bláber. – I buy blueberries.

In your sentence, the natural interpretation is blueberries (plural).

What does the ending -inni in búðinni tell us?

The word búðinni comes from the noun búð (shop).

The ending -inni shows three things at once:

Case: dative
Number: singular
Definiteness: the shop (not just any shop)

So:

basic noun: búð – a shop
definite nominative: búðin – the shop (as subject)
definite dative: búðinni – in/at the shop

In the sentence, the preposition í (in) requires the dative case for location, so we get í búðinni = in the shop.

Which case does í take here, and how does that affect the meaning?

The preposition í can take dative or accusative, and the choice changes the meaning:

Í + dative = location (being in somewhere)
- Ég er í búðinni. – I am in the shop.
Í + accusative = motion (going into somewhere)
- Ég fer í búðina. – I go into the shop.

In your sentence, í búðinni uses dative (búðinni), so it clearly means in the shop, not into the shop.
That matches the idea that the buying is happening inside the shop.

Why is horninu in that form, and what is á horninu doing in the sentence?

Horninu comes from the noun horn (corner, horn).

basic noun: horn – a corner
definite nominative: hornið – the corner
definite dative: horninu – on/at the corner

The preposition á can also take accusative or dative:

á + dative = location (on/at)
- Ég er á horninu. – I am at the corner.
á + accusative = motion onto
- Ég fer á hornið. – I go onto the corner (or to the corner).

In á horninu, horninu is dative, so it means on the corner / at the corner.

Putting it together:
í búðinni á horninu = in the shop on the corner (a specific shop that is located on that corner).

Why is there no separate word for the in búðinni and horninu?

Icelandic does have a definite article like the, but it is usually attached to the end of the noun instead of being a separate word.

For example:

búð – shop
- búð (indefinite: a shop)
- búðin (nom. def.: the shop)
- búðinni (dat. def.: in the shop)
horn – corner
- horn (indefinite: a corner)
- hornið (nom. def.: the corner)
- horninu (dat. def.: at/on the corner)

So búðinni and horninu each already contain the idea the, so you do not add another separate word for the in front of them.

Could the word order be different, like Ég kaupi í búðinni bláber á horninu?

Basic Icelandic word order is Subject – Verb – (Object) – Other information, similar to English:

Ég kaupi bláber í búðinni á horninu.
I (subject) buy (verb) blueberries (object) in the shop on the corner (place).

You usually keep the object (bláber) before the place-phrase (í búðinni á horninu).

The version Ég kaupi í búðinni bláber á horninu sounds unnatural and confusing, because it splits things in a way Icelanders don’t usually do.

You can move the place-phrase to the front for emphasis, but because Icelandic has a verb-second rule, the finite verb must stay in second position:

Í búðinni á horninu kaupi ég bláber.
In the shop on the corner, I buy blueberries. (emphasis on the location)

So: word order can change for emphasis, but it has to follow Icelandic word-order rules, and your original sentence is the neutral, natural version.

Does Ég kaupi mean I buy or I am buying? How does Icelandic show this difference?

The form Ég kaupi can correspond to both English meanings:

I buy blueberries (habitual: I often do this)
I am buying blueberries (right now)

Icelandic present tense does not make a grammatical distinction between simple and continuous like English does. Context normally tells you which is meant.

There is a construction that looks like English continuous:

Ég er að kaupa bláber. – literally I am at to buy blueberries, often used as I am (in the process of) buying blueberries.

You use Ég er að kaupa… when you really want to stress the ongoing process (for example, talking on the phone: I can’t talk, I’m buying something right now).
Otherwise Ég kaupi bláber í búðinni á horninu can naturally be translated as I buy / I am buying blueberries in the shop on the corner, depending on context.

Can I leave out Ég and just say Kaupi bláber í búðinni á horninu?

No. Unlike Spanish or Italian, Icelandic is not a “pronoun-dropping” language in normal sentences.

You must include the subject pronoun:

Ég kaupi bláber í búðinni á horninu. – correct
Kaupi bláber í búðinni á horninu. – ungrammatical as a normal statement

The main exception is imperatives (commands), where you leave out þú:

Kauptu bláber! – Buy blueberries!

But for ordinary statements in the present tense, you keep Ég.

How would the sentence change if I wanted to say into the shop instead of in the shop?

To express into the shop, you need í + accusative (motion), and usually a different verb (go) rather than kaupa (buy):

Ég fer í búðina á horninu til að kaupa bláber.
I go into the shop on the corner to buy blueberries.

Notice the change:

í búðinni – dative (búðinni) → in the shop (location)
í búðina – accusative (búðina) → into the shop (movement)

Using Ég kaupi bláber í búðina á horninu would sound odd, because kaupa describes the act of buying, which normally happens in the shop, not into it. You typically pair í búðina with a verb of motion like fara (to go), hlaupa (to run), etc., not with kaupa.