Мне лучше лечь пораньше, потому что завтра утром экзамен.

Because this is a very common Russian pattern:

кому-то лучше + infinitive

Literally, Мне лучше лечь пораньше is something like For me, it is better to go to bed earlier.

So мне is in the dative case, not the nominative. Russian often uses this kind of impersonal structure where English would simply say I’d better... or I should...

Compare:

• Мне лучше уйти. = I’d better leave.
• Тебе лучше отдохнуть. = You’d better rest.

So мне here means for me / to me, not me as a direct object.

Here лучше means better in the sense of it would be better to...

So:

• Мне лучше лечь пораньше = I’d better go to bed early
• more literally: It would be better for me to go to bed earlier

Grammatically, лучше is the comparative form of хорошо and is also related to лучший (better / best), but in this sentence it functions like an adverb/predicate meaning better.

A useful pattern to remember is:

• лучше + infinitive
• кому-то лучше + infinitive

Examples:

• Лучше подождать. = It’s better to wait.
• Нам лучше поторопиться. = We’d better hurry.

Because лечь is perfective, and here the speaker means a single completed action: going to bed at an earlier time tonight.

• лечь = to lie down / go to bed once, as a completed act
• ложиться = to be lying down / to go to bed habitually or as a process

So:

• Мне лучше лечь пораньше = I’d better go to bed early tonight.
• Мне лучше ложиться пораньше = I should go to bed earlier in general / as a habit.

That perfective vs. imperfective difference is very important in Russian.

Strictly speaking, лечь literally means to lie down. But in the right context, it very often means to go to bed.

In this sentence, because of пораньше and because there’s an exam tomorrow morning, the natural interpretation is:

I’d better go to bed early.

If you want to make sleep completely explicit, you can say:

• Мне лучше лечь спать пораньше.

That is even closer to I’d better go to sleep / go to bed early.

So yes, лечь alone can imply go to bed when the context makes that obvious.

Пораньше means a bit earlier or somewhat early. It often sounds natural and conversational in advice like this.

• раньше = earlier / before
• пораньше = a little earlier, nice and early

So:

• лечь раньше = go to bed earlier
• лечь пораньше = go to bed a bit earlier / go to bed early-ish

In everyday speech, пораньше is very common when someone means earlier than usual in a practical, non-precise way.

Also, раньше can mean before / in the past, so пораньше is often clearer for this kind of sentence.

Because утром is the standard adverbial form meaning in the morning.

It is historically the instrumental case, but learners often just remember these as fixed time expressions:

• утром = in the morning
• днём = in the daytime / during the day
• вечером = in the evening
• ночью = at night

So:

• завтра утром = tomorrow morning

You do not normally say завтра утро for this meaning.

Because Russian usually omits the verb “to be” in the present tense.

So:

• завтра утром экзамен literally = tomorrow morning [is] exam
• natural English = there’s an exam tomorrow morning or I have an exam tomorrow morning

Russian does this all the time:

• Он дома. = He is at home.
• Это моя книга. = This is my book.
• Завтра экзамен. = There is an exam tomorrow.

If the speaker wants to make I have explicit, they could say:

• У меня завтра утром экзамен.

But in your sentence, the shorter version is perfectly natural.

Russian has no articles, so экзамен can mean:

• an exam
• the exam

The exact meaning depends on context.

In English, this sentence is often translated naturally as:

• I’d better go to bed early, because I have an exam tomorrow morning.

Even though the Russian literally just says tomorrow morning [there is] exam, English often prefers I have an exam in this situation.

So the lack of a / an / the is normal in Russian.

No, Russian word order is fairly flexible. The version you have is natural and neutral:

Мне лучше лечь пораньше, потому что завтра утром экзамен.

This puts the time expression first in the second clause:

• завтра утром = tomorrow morning
• экзамен = exam

That sounds very natural because it sets the time first.

Other orders are possible, but they shift emphasis:

• ...потому что экзамен завтра утром.
  This emphasizes exam a bit more.

• ...потому что у меня завтра утром экзамен.
  This makes I have an exam explicit.

So the sentence is not rigidly fixed, but the original version is a very normal one.

Because потому что introduces a subordinate clause meaning because.

Russian normally places a comma before it:

• Я ушёл, потому что устал. = I left because I was tired.
• Мне лучше лечь пораньше, потому что завтра утром экзамен.

So the comma rule here is very straightforward: when потому что introduces the reason clause, you normally put a comma before it.

Yes — absolutely. That is very natural.

Compare:

• Мне лучше лечь пораньше. = I’d better go to bed early.
• Мне лучше лечь спать пораньше. = I’d better go to sleep / go to bed early.

The version with спать is a little more explicit. The shorter version is also very common because the meaning is already obvious from context.

So both are good; the shorter sentence just sounds slightly more compact.

A native speaker would understand it as something like:

I’d better go to bed early, because I have an exam tomorrow morning.

What makes it sound natural in Russian is the combination of:

• мне лучше = I’d better / it would be better for me
• лечь = go to bed / lie down
• пораньше = a bit earlier than usual
• завтра утром экзамен = there’s an exam tomorrow morning

So even though the structure is not word-for-word the same as English, the overall feeling is very idiomatic and natural in Russian.