Learn Hardcore Croatian: On pjeva pjesmu tihim glasom i drži mobitel u ruci. - He is singing a song in a quiet voice and holding his phone in his hand.

On pjeva pjesmu… = He is singing a song… (neutral, or with slight emphasis on he as opposed to someone else).
Pjeva pjesmu… = [He/She/Someone] is singing a song… (subject is clear from context or not important).

So yes, Pjeva pjesmu tihim glasom i drži mobitel u ruci. is also grammatically correct; you include On only if you want to make the subject explicit or contrastive.

What tense and form is pjeva, and how is the verb pjevati conjugated?

Pjeva is:

present tense
3rd person singular
of the verb pjevati (to sing), which is imperfective (focuses on the ongoing nature of the action).

Present-tense conjugation of pjevati:

ja pjevam – I sing / I am singing
ti pjevaš – you sing (sg.)
on/ona/ono pjeva – he/she/it sings
mi pjevamo – we sing
vi pjevate – you sing (pl./formal)
oni/one/ona pjevaju – they sing

So On pjeva literally means He sings / He is singing.

Why is it pjesmu and not pjesma? What case is that?

Pjesma is a feminine noun meaning song. Its nominative singular (dictionary form) is pjesma, but here it is the direct object of pjeva (what is he singing?), so it must be in the accusative case.

For a typical feminine noun in -a, the forms are:

nominative: pjesma – subject (The song is nice.)
accusative: pjesmu – direct object (He sings a song.)

So pjeva pjesmu = he is singing a song (using the accusative pjesmu).

Is it necessary to say pjeva pjesmu, or could you just say On pjeva?

You can absolutely just say On pjeva. That already means He is singing.

Adding pjesmu answers the implicit question What is he singing?

On pjeva. – He is singing (activity in general; content not specified).
On pjeva pjesmu. – He is singing a song (explicit object).

In everyday speech, both are natural; you choose based on whether the object matters in context.

What does the phrase tihim glasom literally mean, and what case is it?

Tihim glasom literally means with a quiet voice or in a quiet voice.

Grammatically:

tihim – instrumental singular of the adjective tih (quiet), masculine
glasom – instrumental singular of glas (voice), masculine

So tihim glasom is in the instrumental case, which is commonly used for:

means: with a knife, with a pen
manner: with a quiet voice, with enthusiasm

Here it expresses the manner/means of singing: He sings a song with a quiet voice.

Why is the adjective tihim in that form? How does it agree with glasom?

In Croatian, adjectives agree with the nouns they describe in gender, number, and case.

glas is masculine singular. In the instrumental case, it becomes glasom.
A masculine singular adjective in instrumental ends in -im (for hard stems like tih), so tih becomes tihim.

Thus:

nominative: tih glas – a quiet voice
instrumental: tihim glasom – with a quiet voice

Both parts are singular, masculine, instrumental, so they match correctly.

Could you say On pjeva pjesmu tiho instead of tihim glasom? Is there any difference?

Yes, you can say On pjeva pjesmu tiho i drži mobitel u ruci. It is perfectly correct.

tiho is an adverb meaning quietly.
tihim glasom is literally with a quiet voice.

In practice:

tiho is a simpler, more neutral way to say he is singing quietly.
tihim glasom can sound a bit more descriptive or stylistic, explicitly mentioning the voice as the instrument of singing.

Meaning-wise, they are very close in this sentence.

What does u ruci mean grammatically, and why is it ruci, not ruka?

U ruci literally means in (the) hand.

ruka is a feminine noun (hand, arm).
After the preposition u meaning “in/inside” (location), Croatian uses the locative case.
Locative singular of ruka is ruci.

So:

nominative: ruka – the hand (subject)
locative: u ruci – in (his/her/the) hand

That’s why the sentence has u ruci, not u ruka.

What would be the difference between u ruci and u ruku?

u ruci – locative case: in the hand (static location)
u ruku – accusative case: into the hand (motion towards)

Examples:

Drži mobitel u ruci. – He is holding the phone in his hand (already there).
Stavi mobitel u ruku. – Put the phone into your hand (movement).

In your sentence, there is no movement, just a state, so u ruci is correct.

How flexible is the word order here? Can I move tihim glasom or u ruci around?

Word order in Croatian is more flexible than in English. You can rearrange parts for emphasis, while the basic meaning stays the same. For example:

On tihim glasom pjeva pjesmu i drži mobitel u ruci.
On pjeva pjesmu tihim glasom i u ruci drži mobitel.
Pjesmu pjeva tihim glasom i drži mobitel u ruci.

All are grammatical. The neutral, most common spoken order is very close to your original sentence. Moving elements usually changes what is stressed or contrasted, not the core meaning.

What exactly does mobitel mean, and is it the usual Croatian word for “phone”?

Mobitel is the standard Croatian word for a mobile phone / cell phone.

It is masculine: taj mobitel, vidim mobitel, s mobitelom.
In other varieties of the language you might hear mobilni telefon, telefon, or mobilni as shorthand.

In everyday Croatian in Croatia, mobitel is the normal, very common word.

Why is the verb drži used with mobitel? Could you also say ima mobitel u ruci?

držati = to hold
imati = to have, to possess

Drži mobitel u ruci. focuses on the physical action and position:

He is holding the phone in his hand.

Ima mobitel u ruci. is grammatically possible and understandable, but it sounds more like He has a phone in his hand (stating that there is a phone in his hand, less about the active holding).

In this context, drži is the natural, idiomatic choice to describe how he is physically holding the phone.