Der Akku meines Handys ist leer.

Because in German, every noun has its own grammatical gender, and that gender does not depend on the gender of another noun nearby.

• der Akku = masculine
• das Handy = neuter

So even though the battery belongs to the phone, Akku keeps its own gender, which is why the sentence starts with Der Akku.

Because German is using the genitive case here to show possession: the battery of my phone / my phone’s battery.

The structure is:

• der Akku = the battery
• meines Handys = of my phone

So meines Handys is a genitive phrase modifying Akku.

Why the forms look like that:

• mein becomes meines because Handy is neuter and in the genitive singular
• Handy becomes Handys because many neuter nouns, especially loanwords, add -s in the genitive

So:

• nominative: mein Handy
• genitive: meines Handys

This is the normal genitive singular ending for many German nouns, especially neuter nouns and many borrowed words.

• das Handy
• des Handys

In this sentence, the article/pronoun is replaced by meines, so you get:

• meines Handys

Important: there is no apostrophe in standard German here.
So it is Handys, not Handy’s.

Because leer is a predicate adjective here, not an adjective directly in front of a noun.

After verbs like:

• sein = to be
• werden = to become
• bleiben = to remain

adjectives usually do not take endings.

So:

• Der Akku ist leer.
• Das Handy ist neu.
• Der Bildschirm bleibt dunkel.

But when the adjective comes before a noun, it does take an ending:

• der leere Akku
• ein leerer Akku

So the sentence uses ist leer, which is why the adjective stays plain.

The structure is:

• Der Akku meines Handys = subject
• ist = verb
• leer = predicate adjective

So the full pattern is:

Subject + verb + adjective

More specifically:

• Der Akku = the battery
• meines Handys = of my phone
• ist leer = is empty / is dead

German main clauses normally put the finite verb in second position, and that is exactly what happens here with ist.

Yes, leer literally means empty, but in German it is also the normal word for a battery that has no charge left.

So for a battery, leer often corresponds to English:

• dead
• flat
• out of charge

This is very natural German:

• Der Akku ist leer.
• Mein Handy ist leer. (informal, understood as “the battery is dead”)

In German, das Handy means mobile phone / cell phone.

This is a famous false friend for English speakers, because English handy is usually an adjective meaning useful or convenient. In German, however, Handy is a noun.

So:

• German das Handy = phone
• English handy ≠ phone

Not exactly.

• der Akku usually means a rechargeable battery
• die Batterie usually means a non-rechargeable battery, though in everyday speech people are not always perfectly strict

For a mobile phone, Akku is the most natural word, because phone batteries are rechargeable.

So Der Akku meines Handys ist leer sounds very normal.

Yes, you can. That is a very common spoken alternative.

Compare:

• Der Akku meines Handys ist leer.
  More compact, a bit more formal or written in style.

• Der Akku von meinem Handy ist leer.
  Very common in everyday speech.

Both are natural. German often uses von + dative instead of the genitive in conversation.

Yes. Mein Handy-Akku ist leer is also correct and very natural.

This uses a compound noun:

• Handy-Akku = phone battery

German often prefers compounds, so this version may even feel especially straightforward.

Compare:

• Der Akku meines Handys ist leer.
• Mein Handy-Akku ist leer.

Both mean essentially the same thing.

Because the subject is singular:

• der Akku = the battery

So the verb must agree with that singular subject:

• Der Akku ... ist leer.

If the subject were plural, the verb would change:

• Die Akkus meiner Handys sind leer.

So ist matches the singular noun Akku.

Normally, no. In standard German, a singular countable noun like Akku usually needs some kind of determiner here:

• Der Akku meines Handys ist leer.
• Mein Handy-Akku ist leer.
• Ein Akku ist leer.

Leaving out the article in this sentence would sound incomplete in normal standard German.

A rough English-friendly guide is:

• meines ≈ MY-ness
• Handys ≈ HAN-dees

A few notes:

• ei in meines sounds like English eye
• y in Handy is pronounced like a long ee
• the final -s in Handys is pronounced like an s/z sound, not as a separate syllable

So the whole sentence is roughly:

dare AH-koo MY-ness HAN-dees ist lair

That is only an approximation, but it helps as a starting point.