Learn Hardcore Russian: Мой будильник звенит утром.

In this sentence, будильник is the subject of the verb звенит (it is the alarm that does the ringing), and subjects normally appear in the nominative case in Russian.
So: Мой будильник = My alarm clock (subject).

What exactly does звенит mean? Is it rings or is ringing?

Звенит is the 3rd person singular present tense of звенеть (to ring, to chime, to jingle).

Russian does not have a separate continuous form like English is ringing.
Depending on context, звенит can mean:

rings (habitual, general fact):
- Мой будильник звенит утром. = My alarm rings in the morning.
is ringing (right now), if the situation makes it clear:
- Будильник звенит! = The alarm is ringing!

So звенит can cover both rings and is ringing; context decides.

What is the infinitive of звенит, and how is this verb conjugated?

The infinitive is звенеть (to ring).

Present tense conjugation (imperfective):

я звеню – I ring
ты звенишь – you ring (singular, informal)
он / она / оно звенит – he / she / it rings
мы звеним – we ring
вы звените – you ring (plural or formal)
они звенят – they ring

In the sentence, the subject is мой будильник (3rd person singular), so we use звенит.

Why is it утром without a preposition, and not в утро?

Утром is the instrumental form of утро (morning), but in practice it functions as an adverbial expression meaning in the morning / in the mornings.

Russian very often uses the instrumental case of time words without a preposition to say when something happens:

утром – in the morning
днём – in the afternoon / during the day
вечером – in the evening
ночью – at night

You don’t say в утро in this sense. The natural way is simply утром.

Does утром mean every morning, or just this morning?

On its own, утром is somewhat general and usually understood as a regular or typical time when combined with a present-tense habitual verb:

Мой будильник звенит утром.
Normally understood as: My alarm (typically) rings in the morning.

If you want to emphasize every morning, you can say:

Мой будильник звенит каждое утро. – My alarm rings every morning.
Мой будильник по утрам звенит. – My alarm rings in the mornings (habitually).

Can I move утром to the beginning: Утром мой будильник звенит? Is that correct?

Yes, it is correct: Утром мой будильник звенит.

Russian word order is flexible. Both:

Мой будильник звенит утром.
Утром мой будильник звенит.

are grammatically fine.
The second version puts a bit more emphasis on the time (on утром), like In the morning, my alarm rings. But the basic meaning is the same.

Can I drop мой and just say Будильник звенит утром?

Yes, Будильник звенит утром is grammatically correct and natural.

Whether you include мой depends on what you want to emphasize and what is already clear from context:

Мой будильник звенит утром. – Emphasizes it’s my alarm.
Будильник звенит утром. – A neutral statement: The alarm rings in the morning, and the owner may be obvious from context or not important.

In Russian, possessive pronouns are often omitted when ownership is obvious.

How would I say My alarm rang in the morning and My alarm will ring in the morning?

You can keep будильник and утром, and change the verb form and sometimes aspect.

Past (rang):
- Мой будильник звонил утром. – My alarm rang in the morning / was ringing in the morning (general or repeated in the past).
- Мой будильник прозвенел утром. – My alarm rang in the morning (one specific completed event).
Future (will ring):
Imperfective (process / regular action):
- Мой будильник будет звонить утром. – My alarm will be ringing in the morning / will ring (as a regular arrangement).
Perfective (one specific occurrence):
- Мой будильник прозвенит утром. – My alarm will ring in the morning (once, at some point in the future).

In your original habitual sense, будет звонить утром fits best for a future regular arrangement.

Is будильник only for a physical alarm clock, or can it mean a phone alarm too?

Будильник can refer to:

a traditional alarm clock device
an alarm function on a phone or other device (by extension)

For example, Russians naturally say:

Я поставил будильник на телефоне. – I set an alarm on my phone.

So будильник is any alarm you set to wake you up, not only the old-style clock.

What gender is будильник, and how do I know?

Будильник is masculine.

Clues:

It ends in a consonant (-к), which is the most common pattern for masculine nouns.
The possessive pronoun мой also shows it is masculine, because мой is used only with masculine singular nouns in nominative.

Gender affects:

the choice of pronouns and modifiers: мой будильник, старый будильник
agreement in past tense:
- Будильник прозвенел. (masculine past tense form)

How do you pronounce and stress each word in this sentence?

Approximate pronunciation with stressed syllables in capitals:

Мой – [MOY], like moy in boy but with m
будильник – [boo-DÉEL-neek]
- stress on ди́ль: буди́льник
звенит – [zvee-NEET]
- stress on нит: звени́т
утром – [OO-trum]
- stress on у́: у́тром

So: МОЙ буДИ́льник звеНИ́т У́тром (with actual stress on the bolded syllables).

How would I say My alarm rings every morning more explicitly?

You can add words that clearly show repetition:

Мой будильник звенит каждое утро. – My alarm rings every morning.
Мой будильник по утрам звенит. – My alarm rings in the mornings (habitually).

Both emphasize the regular, repeated nature of the action more strongly than just утром.