Learn Hardcore Ukrainian: Мені трохи страшно уночі.

This is called an impersonal construction. There is no clear “I” as the grammatical subject of a verb; instead, the feeling “is scary” exists for me.

Similar patterns:

Мені холодно. – I’m cold. (Literally: To me it’s cold.)
Тобі сумно. – You feel sad. (To you it’s sad.)
Нам нудно. – We are bored. (To us it’s boring.)

Using Я here (Я страшно уночі) is ungrammatical in this meaning.

What exactly is страшно? Is it an adjective, an adverb, or something else?

Страшно comes from the adjective страшний (scary, terrible), but in this sentence it functions as a predicative adverb (українською: предикативний прислівник).

Key points:

It does not agree with any noun in gender, number, etc.
It behaves like a fixed form used to describe a state: scary, cold, sad, etc.
Ukrainian has a whole group of such words:
холодно, жарко, боляче, сумно, приємно, соромно, цікаво.

In sentences like Мені страшно, страшно means something like “it feels scary” or “I feel scared”, not “I am a scary person”.

What does трохи do here, and where can it go in the sentence?

Трохи means “a little, a bit, somewhat.”

In Мені трохи страшно уночі, it softens the feeling: not very scared, just a bit.
It modifies страшно: “a bit scary / a bit scared.”

Typical position is before the word it modifies:

Мені трохи страшно.
Мені трохи холодно.

You might also hear Мені страшно трохи, but that sounds less natural and may shift a slight focus onto “only a bit”. The most neutral version is the one you have: Мені трохи страшно уночі.

Why is there no verb like “to be” (є) in this sentence?

In the present tense, Ukrainian usually drops the verb “to be” (є) in sentences like this.

Full logical structure would be: Мені є трохи страшно уночі.
But є sounds unnatural here, so you almost always say:
Мені трохи страшно уночі.

However, in past and future you must use the verb:

Мені було страшно уночі. – I was scared at night.
Мені буде страшно уночі. – I will be scared at night.

What case is Мені, and how is it different from мене?

Мені is dative (кому? – to whom?): used for experiencer of a feeling/state.
Мене is genitive/accusative (кого? – whom?): used as an object, or after some prepositions.

Compare:

Мені страшно. – I am scared. (To me it’s scary.)
Він боїться мене. – He is afraid of me. (I am the object of his fear.)

So for personal feelings expressed with words like страшно, холодно, сумно, you use the dative: мені, тобі, йому, нам, etc.

Can I say Я страшний уночі to mean “I’m scared at night”?

No. Я страшний уночі means “I am scary at night” (other people are afraid of me).

страшний = scary, frightening (describes someone/something that causes fear).
страшно (мені страшно) = it is scary (for me) → I feel scared.

So:

Мені трохи страшно уночі. – I feel a bit scared at night.
Я страшний уночі. – I’m a scary person at night. (Very different meaning!)

Is уночі the same as вночі? Which one should I use?

Yes, уночі and вночі are essentially synonyms: both mean “at night / during the night.”

Both are very common and correct.
Many speakers choose у or в mainly for sound reasons (to avoid awkward consonant clusters, to make speech flow more smoothly).

In your sentence you can say:

Мені трохи страшно уночі.
Мені трохи страшно вночі.

Both sound natural. You don’t change the meaning.

What is the form of уночі? Is it a preposition plus a noun?

Historically, уночі comes from у + ночі (in the night, at night: preposition у + locative case of ніч).

In modern Ukrainian, уночі / вночі are often treated as adverbs of time, similar to:

увечері / ввечері – in the evening
вранці / уранці – in the morning

So you can think of уночі as one fixed time expression: “at night.” You don’t need to worry about its internal structure to use it correctly.

Can I change the word order, for example: Уночі мені трохи страшно?

Yes, Ukrainian word order is quite flexible, and your example is natural:

Уночі мені трохи страшно. – perfectly fine, just puts a bit more emphasis on “at night” (contrast: at night vs during the day).
Мені трохи страшно уночі. – more neutral order; the focus is on the feeling.

Other natural variants:

Мені уночі трохи страшно.
Мені трохи страшно вночі.

The basic meaning stays the same; you mainly shift what part of the sentence feels more prominent.

What’s the difference between Мені страшно and Я боюся?

Both can describe fear, but they feel a bit different:

Мені страшно.
- Focus on your inner state: “I feel scared / it is scary (for me).”
- Often more emotional, subjective, spontaneous.
- Very common in everyday speech.
Я боюся.
- кого? чого? / за кого?
- Uses the verb боятися (to fear, to be afraid).
- Often needs an object:
  - Я боюся темряви. – I’m afraid of the dark.
  - Я боюся за тебе. – I’m worried/afraid for you.
- Sounds a bit more explicit and can be more concrete.

You could easily say:

Мені трохи страшно уночі, бо я боюся темряви.
→ I’m a bit scared at night because I’m afraid of the dark.