Otkad sam preuzela tu aplikaciju, mogu brže preuzeti i velike datoteke.

Here otkad means since or ever since. It introduces a time clause:

• Otkad sam preuzela tu aplikaciju... = Since I downloaded that app...

A very close variant is otkako, which means the same thing. Both are common.

Croatian usually forms the past tense with:

• the present tense of biti (to be)
• plus a past participle

So:

• sam = I am / auxiliary for I have
• preuzela = past participle of preuzeti

Together, sam preuzela means I downloaded / I have downloaded.

This is the normal Croatian perfect tense.

Because the speaker is grammatically feminine.

In Croatian, the past participle agrees with the gender of the speaker in the singular:

• preuzeo = a male speaker
• preuzela = a female speaker

So a man would say:

• Otkad sam preuzeo tu aplikaciju...

Because sam is a clitic, and Croatian clitics usually go very early in the clause, often in second position.

So:

• Otkad sam preuzela tu aplikaciju is the natural standard order.

This kind of placement is very common in Croatian:

• Kad sam došla...
• Ako se sjećaš...
• Jer sam mislio...

Because preuzeti takes a direct object, and the direct object here is in the accusative case.

The base form is:

• ta aplikacija = that app (nominative)

But as the object of preuzela, it becomes accusative:

• tu aplikaciju

Both the demonstrative and the noun change to match the accusative feminine singular.

Yes, here tu means that.

It is the accusative feminine singular form of the demonstrative taj / ta / to:

• ta aplikacija = that app
• tu aplikaciju = that app as a direct object

You are right that tu can also be an adverb meaning there or sometimes here in certain contexts, but here it is clearly a demonstrative because it directly modifies aplikaciju.

They are two different forms of the same verb:

• preuzela = past participle, used in the past tense
• preuzeti = infinitive

So the sentence contains:

1. a completed past action

   • sam preuzela = I downloaded

2. a present ability

   • mogu preuzeti = I can download

This is very normal in Croatian:

• Kupila sam auto i sad mogu brže putovati.
• Instalirao sam program i sad mogu raditi lakše.

Because the sentence means that the past action has a result that is true now.

So:

• Otkad sam preuzela tu aplikaciju = a past event
• mogu brže preuzeti i velike datoteke = my current ability/result

In English, this is similar to:

• Ever since I downloaded that app, I can download even large files faster.

The past event happened once, but the improved ability continues into the present.

Because mogu preuzeti means I can download or I am able to download.

The verb moći expresses ability or possibility:

• mogu = I can

Then it is followed by an infinitive:

• mogu preuzeti = I can download

If you said preuzimam, that would mean I am downloading or I download, which is a different idea.

Good question. In Croatian, aspect matters.

• preuzeti is perfective: it focuses on completing the download
• preuzimati is imperfective: it focuses on the process or repeated action

Here mogu brže preuzeti i velike datoteke suggests successful, completed downloads of even large files, so the perfective verb fits well.

If you used the imperfective, the meaning would shift a little more toward the ongoing process or habitual activity.

Brže means faster.

It is the comparative form of the adverb brzo (fast, quickly):

• brzo = fast / quickly
• brže = faster / more quickly

So:

• mogu brže preuzeti = I can download faster

Croatian often forms adverb comparatives this way:

• sporo → sporije
• lako → lakše
• brzo → brže

Here i is best understood as even or also.

• i velike datoteke = even large files / large files too

The idea is that the app improves downloading so much that not only ordinary files, but even large ones, can be downloaded faster.

So although i often means and, in this sentence it has an emphatic meaning closer to also/even.

It is the accusative plural, because it is the direct object of preuzeti.

Base form:

• velike datoteke can also be nominative plural

Here it functions as accusative plural:

• preuzeti velike datoteke = to download large files

For feminine plural inanimate nouns like datoteke, the nominative and accusative plural often look the same, so the form does not visibly change.

Yes, often you can.

• preuzeti is a standard, neutral verb for download
• skinuti is also very common in everyday speech for downloading something from the internet

So a native speaker might also say something like:

• Otkad sam skinula tu aplikaciju...

But preuzeti is a very good standard choice, especially in careful or neutral language.