Baterija na mobitelu se sasvim ispraznila.

Because it comes after na, which here requires the locative case. The locative singular of mobitel is mobitelu.

So:

• mobitel = nominative
• na mobitelu = on / in / on the phone, with mobitelu in the locative

Yes, literally it is on the phone, but Croatian often uses na with devices in a broader way than English does. It can refer to something associated with the device, not only something physically sitting on top of it.

So baterija na mobitelu is a natural everyday way to refer to the phone’s battery. Depending on context, you may also hear:

• baterija u mobitelu = the battery in the phone
• baterija mobitela = the phone’s battery

These are all possible, but they differ a bit in style:

• na mobitelu = everyday, conversational
• u mobitelu = more physical/literal
• mobitela = more formal or technical

Here se is part of the verb isprazniti se. It does not mean itself in a literal English sense. Instead, it helps make the verb intransitive:

• isprazniti nešto = to empty something / drain something
• isprazniti se = to become empty / to drain completely

So the sentence is not saying the battery emptied itself in a deliberate sense. It just means the battery became drained.

Normally, the Croatian perfect does use an auxiliary:

• on je došao
• ona je radila

But with many reflexive verbs in the 3rd person singular, Croatian normally leaves out je:

• Ona se vratila
• Baterija se ispraznila

So Baterija na mobitelu se sasvim ispraznila is the normal form. Learners often expect je, but here its absence is standard and natural.

Because the past participle agrees with baterija, which is feminine singular.

Croatian past forms show gender and number:

• masculine singular: ispraznio se
• feminine singular: ispraznila se
• neuter singular: ispraznilo se
• plural: ispraznili / ispraznile se

Since baterija is feminine, the form must be ispraznila.

This is a question of aspect.

• prazniti se = imperfective, ongoing/repeated process
• isprazniti se = perfective, completed result

In this sentence, the idea is that the battery reached the result of being drained, so Croatian uses the perfective verb isprazniti se.

Compare:

• Baterija se praznila. = The battery was draining / kept draining.
• Baterija se ispraznila. = The battery drained completely / ran out.

The prefix is- often adds the idea of completion.

Sasvim means completely, entirely, or totally here. It strengthens the idea that the battery is not just low, but fully drained.

Without it:

• Baterija na mobitelu se ispraznila. = The battery drained / ran out.

With it:

• Baterija na mobitelu se sasvim ispraznila. = The battery completely drained.

Also note that sasvim can mean quite in some contexts, so its exact shade depends on the sentence.

Because se is a clitic. Croatian clitics usually appear very early in the sentence, typically after the first phrase or near the beginning of the clause.

So in:

Baterija na mobitelu se sasvim ispraznila

the clitic se comes early, before the main verb. This is normal Croatian word order. English speakers often want to place it more freely, but Croatian clitics have stricter placement rules.

Yes. A few natural variants are:

• Baterija na mobitelu se ispraznila.
  Same idea, just without sasvim.

• Baterija u mobitelu se sasvim ispraznila.
  More literally the battery in the phone.

• Ispraznila mi se baterija na mobitelu.
  Very natural in speech; mi adds the idea that it happened to me / my battery died.

• Mobitel mi se ispraznio.
  Very common conversational shortcut meaning the phone ran out of battery.

So the original sentence is perfectly natural, but Croatian offers several equally common ways to express the same situation.