Learn Hardcore Ukrainian: На уроці вчителька пояснює нове правило, і мені спочатку складно. - In class the teacher explains a new rule, and at first it is difficult for me.

After the preposition на, when it means location or time, many masculine nouns answer "де?" / "коли?" (where? / when?) and go into the prepositional case:

на столі (on the table)
на змаганні (at the competition)
на уроці (in class)

What is the difference between вчителька and вчитель?

Both mean teacher, but they differ in gender:

вчитель – male teacher (grammatically masculine)
вчителька – female teacher (grammatically feminine)

So вчителька tells you the teacher is a woman. Grammatically:

вчитель: nominative singular masculine
вчителька: nominative singular feminine

Why is пояснює used here, and what tense/aspect is it?

Пояснює is 3rd person singular, present tense, imperfective of пояснювати (to explain).

It can correspond to both:

"explains" (general, habitual)
"is explaining" (right now)

Ukrainian has no separate present continuous form, so context decides whether you read it as "explains" or "is explaining".

What is the difference between пояснює and пояснить?

пояснює – imperfective, present tense of пояснювати
- Focus on the process of explaining or repeated action
- Вчителька пояснює правило. – The teacher is explaining / explains the rule.
пояснить – perfective, future tense of пояснити
- Focus on the result: the explanation will be completed
- Вчителька пояснить правило. – The teacher will explain the rule (and then you will know it).

In the sentence вчителька пояснює нове правило, we are looking at what is happening during the lesson (process), so imperfective пояснює fits.

Why is it нове правило and not новий правило?

Because правило is neuter, and adjectives must agree in gender, number, and case.

правило – neuter, singular, nominative/accusative
The neuter singular form of новий is нове.

So:

masculine: новий стіл (new table)
feminine: нова книжка (new book)
neuter: нове правило (new rule)

What case is нове правило in, and why?

Нове правило is in the accusative case as the direct object of пояснює.

Verb: пояснює (explains)
Direct object: що? (what?) → нове правило

For neuter nouns like правило, the accusative form is identical to the nominative:

Nominative: нове правило
Accusative: нове правило

Why is there a comma before і in ..., і мені спочатку складно?

In Ukrainian, a comma is usually placed before і when it connects two independent clauses (two separate sentences that could stand alone).

Here we have:

На уроці вчителька пояснює нове правило – full sentence
мені спочатку складно – another full sentence (impersonal structure)

They are joined by і, so a comma is used:

На уроці вчителька пояснює нове правило, і мені спочатку складно.

Why is it мені and not я or мене?

Мені is dative case of я.

Forms of я:

Nominative: я (I)
Accusative/Genitive: мене (me)
Dative: мені (to me / for me)

In Ukrainian, feelings, states, and experiences are often expressed with dative + predicate:

Мені жарко. – I am hot. (literally: To me it is hot.)
Мені сумно. – I am sad.
Мені складно. – It is difficult for me / I find it hard.

So мені спочатку складно literally means "To me, at first, (it is) difficult."

Why is there no є in мені спочатку складно? Should it be мені спочатку є складно?

In the present tense, Ukrainian usually drops the verb "to be" (є) in such sentences.

Full theoretical form: Мені спочатку є складно.
Natural, normal Ukrainian: Мені спочатку складно.

In predicate constructions like X є Y, in the present tense the є is normally omitted:

Я студент. – I am a student.
Вона вчителька. – She is a teacher.
Мені складно. – It is difficult for me.

You use є mainly for emphasis or in some formal/written styles.

What is складно grammatically? Is it an adjective or an adverb?

Formally, складно is an adverbial form (from the adjective складний – difficult), but in this type of sentence it functions as a predicative word (part of the sentence that says what the situation is like).

It is used in impersonal constructions describing how something feels:

Мені складно. – It is difficult for me.
Тобі легко. – It is easy for you.
Йому страшно. – He is scared / It is scary for him.

So you don’t say я складний here; you use мені складно.

What does спочатку mean exactly, and are there synonyms?

Спочатку means "at first / in the beginning".

Common synonyms:

спершу – at first
на початку – at the beginning (often followed by чогось: на початку уроку – at the beginning of the lesson)

You can use:

Мені спочатку складно. – It is difficult for me at first.
Мені спершу складно. – Same meaning, more colloquial.

Can the word order be changed, for example: Вчителька на уроці пояснює нове правило or Спочатку мені складно?

Yes, Ukrainian word order is fairly flexible, and all of these are possible:

На уроці вчителька пояснює нове правило...
– Slight focus on "during the lesson" (setting first).
Вчителька на уроці пояснює нове правило...
– Slight focus on "the teacher" first, then her location.
Мені спочатку складно. (as in the original)
– Neutral emphasis.
Спочатку мені складно.
– Emphasis on "at first" – contrast with what happens later.

All of these are grammatical; the difference is nuance and emphasis, not correctness.

How is вчителька pronounced, and is the вч cluster normal?

Вчителька is pronounced approximately as [VCHY-tel-ka]:

вч – a consonant cluster, similar to "vch" in English "kvetch" (but without the k)
Stress falls on the first syllable: ВЧИ-тель-ка

Consonant clusters like вч, вступ, втрата are normal in Ukrainian, even at the beginning of a word.

Why do we say на уроці but в класі? Is there a rule?

Broadly:

Use на + Locative for events / activities / functions:
- на уроці – in class / during the lesson
- на роботі – at work
- на концерті – at a concert
Use в / у + Locative for physical locations, enclosed spaces:
- в класі – in the classroom
- в кімнаті – in the room
- в будинку – in the house

So:

Я на уроці. – I am in class (attending a lesson).
Я в класі. – I am in the classroom (physically in the room).

In your sentence, the focus is on the activity (the lesson), so на уроці is used.