Будь ласка, говори тихіше, бо дитина спить.

Both verbs can relate to speaking, but говорити is the more natural choice for speak / talk in this kind of request.

• говори тихіше = speak more quietly
• кажи usually means say / tell

So Кажи тихіше would sound less natural here, because the focus is on the manner of speaking, not on saying a particular thing.

Говори is the imperative singular form of говорити.

It is used when telling one person to do something:

• Говори тихіше. = speaking to one person informally

Related forms:

• Говоріть тихіше. = to more than one person, or to one person formally/politely
• Не говори. = negative imperative, singular

Yes, very normal. Будь ласка is the standard polite phrase for please.

In Ukrainian, it commonly appears with commands or requests:

• Будь ласка, допоможи.
• Будь ласка, зачекайте.

So Будь ласка, говори тихіше is a polite request, not a harsh command.

Тихіше is the comparative form of the adverb тихо.

• тихо = quietly
• тихіше = more quietly / quieter

In requests like this, Ukrainian often uses the comparative where English might say more quietly or simply quieter:

• Говори тихіше. = Speak more quietly / quieter.

This is very common in Ukrainian:

• повільно → повільніше = slowly → more slowly
• швидко → швидше = quickly → faster

Yes, тише is also used and is very common in everyday speech.

Both can mean more quietly / quieter:

• Говори тихіше
• Говори тише

In practice:

• тихіше may sound a bit more clearly connected to тихий
• тише is shorter and very natural conversationally

A learner should recognize both.

Both mean because, but бо is shorter and more conversational.

• бо = common in everyday speech
• тому що = also very common, sometimes slightly fuller or more neutral in style

So these are both natural:

• Будь ласка, говори тихіше, бо дитина спить.
• Будь ласка, говори тихіше, тому що дитина спить.

In casual spoken Ukrainian, бо is especially frequent.

Because Ukrainian normally does not use a verb equivalent to to be in the present tense the way English does.

So:

• дитина спить = the child is sleeping
• literally, it is just child sleeps

This is completely normal in Ukrainian. Present continuous ideas in English are often expressed with a simple present form in Ukrainian, depending on context.

Дитина primarily means child, but in context it can also refer to a baby or young child.

So the sentence could be used if:

• a baby is asleep
• a toddler is asleep
• a child is asleep

If you specifically want baby, Ukrainian also has:

• немовля = infant/baby
• малюк / малеча = little one / little child, depending on context

But дитина is very broad and natural.

Because бо introduces a subordinate clause, and in Ukrainian that clause is normally separated by a comma.

So:

• Будь ласка, говори тихіше, бо дитина спить.

There is also often a comma after Будь ласка, especially when it functions as a parenthetical polite expression at the start of the sentence.

Говоріть тихіше is the correct polite/formal version.

Use it:

• when speaking to one person politely
• when speaking to several people

Examples:

• Будь ласка, говоріть тихіше, бо дитина спить.
• Будь ласка, говоріть тихше.

Кажіть тихіше is less natural for the same reason that кажи тихіше is less natural: казати is more about saying/telling, while говорити is better for speaking in a quieter way.

Спить is the 3rd person singular present form of спати = to sleep.

So:

• я сплю = I sleep / I am sleeping
• ти спиш = you sleep / are sleeping
• він/вона спить = he/she sleeps / is sleeping

In this sentence:

• дитина спить = the child is sleeping

This is one of those verbs where the stem changes a bit across forms, so it is worth memorizing as a pattern.

Not necessarily rude, but less soft.

Compare:

• Говори тихіше. = direct; can sound firm depending on tone
• Будь ласка, говори тихіше. = polite request

As in English, tone and context matter a lot. If you are speaking to a friend or family member, the shorter version may be perfectly fine. But adding Будь ласка makes it clearly polite.