Udělej otočku a otáčej se pomalu.

Because they express two different kinds of action.

• udělej otočku uses a perfective verb idea: do one complete turn/spin
• otáčej se pomalu uses an imperfective verb idea: keep turning / rotate gradually

Czech often distinguishes between:

• a single completed action
• an ongoing or repeated action

So the sentence combines both:

1. first, do a turn
2. then, keep turning slowly

That is a very common Czech pattern.

Udělej is the imperative singular of udělat, which is a perfective verb.

Here that matters because the speaker wants one completed action:
do a turn.

If you said dělej otočku, it would sound less natural in this context, because dělat is imperfective and usually suggests something more ongoing, repeated, or process-focused.

So:

• udělej otočku = do one turn
• dělej otočku = more like keep doing a turn / be doing the turn, which is not usually what is meant here

Otočka is a noun meaning something like:

• a turn
• a spin
• a turnaround

In this sentence, it most likely means one turn/spin.

A useful thing to know is that otočka can also mean different kinds of turning in other contexts, for example:

• a physical spin or turn
• a turnaround
• sometimes even a U-turn in traffic-related language

So the exact nuance depends on context.

Because it is the direct object of udělej, so it has to be in the accusative case.

The base form is:

• ta otočka

But after udělej you need the accusative:

• udělej otočku

This is a normal feminine noun pattern:

• nominative: otočka
• accusative: otočku

This is mostly about aspect again.

• otoč se = turn around / make one completed turn
• otáčej se = keep turning / be turning

So:

• otoč se pomalu would mean turn slowly as one whole action
• otáčej se pomalu means turn/rotate slowly over time

The version in your sentence focuses on the ongoing motion, not just the completed result.

Se is a reflexive pronoun.

In this sentence, it shows that the subject is turning itself:

• otáčet se = to turn / rotate oneself, to turn around

In English we usually do not say turn yourself, because English often leaves that idea unspoken. Czech often expresses it with se.

So:

• otáčej se = turn yourself / keep turning

You will see this a lot with body movement verbs:

• posaď se = sit down
• zastav se = stop
• otoč se = turn around

Yes, you could. Czech word order is fairly flexible.

The original:

• Udělej otočku a otáčej se pomalu.

sounds natural and neutral.

You could also say:

• Udělej otočku a pomalu se otáčej.

That puts a little more emphasis on slowly.

In general:

• otáčej se pomalu = neutral
• pomalu se otáčej = slightly more emphasis on the manner

Both are grammatical.

Yes.

Both imperative forms here are for one person in an informal situation:

• udělej
• otáčej se

If you wanted to speak:

• politely to one person, or
• to several people,

you would use the -te forms:

• Udělejte otočku a otáčejte se pomalu.

So the sentence you have is the ty form.

Yes, very often.

For a simple instruction, otoč se is extremely common and direct.
For example:

• Otoč se a otáčej se pomalu.

However, that changes the nuance a bit:

• udělej otočku treats the first action as a turn/spin
• otoč se focuses more directly on turning yourself around

So your sentence is fine, but a learner should know that otoč se is also very common and often more straightforward in everyday speech.

It sounds like an instruction, for example in:

• dance
• exercise
• physical therapy
• acting/stage movement
• a game or activity involving body movement

The combination of one turn plus slow continued rotation sounds especially natural in a guided physical activity.