Eu tenho dez minutos para descansar.

You can drop eu. In Brazilian Portuguese the subject pronoun is often omitted because the verb ending already shows the person.

• Eu tenho dez minutos para descansar. – a bit more explicit or emphatic.
• Tenho dez minutos para descansar. – perfectly natural in speech and writing when context is clear.

In modern Brazilian Portuguese, ter is the normal verb to express possession or having available time, money, etc. So Eu tenho dez minutos = I have ten minutes (available).
Haver is mostly used in impersonal expressions now (like há dez minutos, ten minutes ago), not for personal possession. Estar would change the meaning to a state, not “having” something, so estou dez minutos is incorrect here.

Para here shows purpose: what those ten minutes are for.

• Eu tenho dez minutos para descansar. – I have ten minutes in order to rest.

Por usually marks duration or cause. You would use por with a verb like this:

• Eu vou descansar por dez minutos. – I’m going to rest for ten minutes.

So:

• para descansar = what the time is for (purpose)
• por dez minutos = how long something lasts (duration)

Yes. Pra is a very common spoken (and informal written) contraction of para in Brazilian Portuguese.

• Eu tenho dez minutos pra descansar. – completely natural in conversation, text messages, etc.
  In formal writing, people prefer para, but pra is everywhere in everyday speech.

Descansar means to rest, to take a rest, to relax. In Brazilian Portuguese it is normally not reflexive here, so:

• Eu preciso descansar. – I need to rest. (not me descansar in Brazil)

It can also be used with an object sometimes (for example, descansar a cabeça – rest your head), but in this sentence it’s just a simple intransitive “to rest.”

No, that word order sounds wrong in Portuguese. Para descansar forms a unit with the verb descansar and normally stays together at the end of the sentence in this structure:

• ✅ Eu tenho dez minutos para descansar.
  Putting para descansar in the middle or in front of dez minutos is unnatural.

The structure stays the same; you just change tenho:

• Past (completed moment):
  
  • Eu tive dez minutos para descansar. – I had ten minutes to rest (on that occasion).
• Past (background / habitual):
  
  • Eu tinha dez minutos para descansar. – I used to / normally had ten minutes to rest.
• Future:
  
  • Eu vou ter dez minutos para descansar. – I’m going to have ten minutes to rest. (most common)
  • Terei dez minutos para descansar. – I will have ten minutes to rest. (more formal/written)

Yes, it’s a very productive pattern. You can replace descansar with many other verbs:

• Eu tenho duas horas para estudar. – I have two hours to study.
• Nós temos pouco tempo para almoçar. – We have little time to have lunch.
• Eles têm um dia inteiro para decidir. – They have a whole day to decide.

Both can be used, but the nuance is slightly different:

• dez minutos para descansar – focuses on the purpose: ten minutes to rest (what you can do with that time).
• dez minutos de descanso – treats descanso as a thing you possess: ten minutes of rest (like having a scheduled “ten‑minute break”).

In many contexts they can feel interchangeable, but para descansar sounds more directly like “time available for resting.”

In Portuguese, when you use a number (other than 1) before a countable noun, the noun is plural:

• um minuto (1 minute)
• dois minutos, dez minutos, trinta minutos (2, 10, 30 minutes)

So after dez, you must use the plural minutos, not the singular form.

Very commonly you’ll hear something like:

• Eu tenho dez minutos pra descansar.

Typical tendencies:

• eu often sounds like a quick “ew”.
• tenho sounds like “TEN-nyo” (the nh is like “ny” in “canyon”).
• para is often shortened to “pra” in speech.
  So in fast, natural speech, it comes out quite compact.

A very natural, slightly more informal version would be:

• Só tenho dez minutos pra descansar. – I only have ten minutes to rest.

You might also hear:

• Tenho só dez minutinhos pra descansar. – I only have ten little minutes to rest. (minutinhos is a friendly/diminutive form, very common in speech.)