Opettaja pyytää meitä vastaamaan kysymyksiin nopeasti.

A quick breakdown:

• opettaja = teacher
  
  • the subject
• pyytää = asks / is asking / requests
  
  • the main verb
• meitä = us
  
  • the person being asked
• vastaamaan = a form of vastata = to answer
  
  • the action being requested
• kysymyksiin = to the questions in Finnish grammar, but naturally just the questions in English
• nopeasti = quickly
  
  • an adverb describing how to answer

So the structure is basically:

teacher + asks + us + to answer + the questions + quickly

Because pyytää uses a special pattern here:

pyytää jotakuta tekemään jotakin
= to ask someone to do something

In that pattern, the person being asked is in the partitive, so:

• minua = me
• sinua = you
• häntä = him/her
• meitä = us
• teitä = you plural/formal
• heitä = them

So:

• Opettaja pyytää meitä... = The teacher asks us...

This is something you mostly learn as a verb pattern.

After pyytää, Finnish usually does not use the basic infinitive vastata here. It uses the -maan / -mään form:

pyytää jotakuta tekemään jotain
= ask someone to do something

So:

• vastata = to answer (dictionary form)
• vastaamaan = the form used after pyytää in this construction

Compare:

• Opettaja pyytää meitä vastaamaan. = The teacher asks us to answer.
• Pyydän sinua odottamaan. = I ask you to wait.

There is no separate word for English to here; Finnish expresses that idea through the verb form -maan / -mään.

It is the third infinitive illative form.

That sounds technical, but the practical point is more important:

• vastata = dictionary form
• vastaamaan = a form often used after verbs like pyytää, and also after some verbs of movement or beginning

A useful learner shortcut is:

• mennä tekemään = go do
• tulla tekemään = come do
• pyytää jotakuta tekemään = ask someone to do

So in this sentence:

• meitä vastaamaan = us to answer

Because vastata works differently from English answer.

In English, answer takes a direct object:

• answer the question

In Finnish, vastata usually takes a case complement instead:

• vastata kysymykseen = answer a question
• vastata kysymyksiin = answer questions
• vastata opettajalle = answer the teacher

So with vastata, you should learn the pattern:

• vastata johonkin = answer something
• vastata jollekin = answer someone

That is why the sentence has kysymyksiin, not a direct object form.

This is just one of those Finnish noun patterns that looks bigger than it feels at first.

The dictionary form is:

• kysymys = question

But the stem changes in other forms:

• kysymyksen = of the question
• kysymyksiä = questions
• kysymyksiin = into/to the questions, used here after vastata

So kysymyksiin is:

• plural
• illative case
• based on a changed stem

This is normal Finnish inflection, even if it looks dramatic at first. It is best to learn words like this with a few key forms, not only the dictionary form.

Nopeasti means quickly.

It comes from the adjective:

• nopea = fast, quick

Finnish often makes adverbs by adding -sti:

• nopea → nopeasti = quickly
• hidas → hitaasti = slowly
• varma → varmasti = certainly / surely

So here nopeasti tells us how the answering should happen.

Not completely. Finnish word order is fairly flexible, because case endings show grammatical relationships.

The sentence as written is neutral and natural:

Opettaja pyytää meitä vastaamaan kysymyksiin nopeasti.

But some movement is possible, especially with the adverb:

• Opettaja pyytää meitä vastaamaan nopeasti kysymyksiin.
• Nopeasti opettaja pyytää meitä vastaamaan kysymyksiin.
  
  • possible, but more marked and less neutral

So the original version is a good default. If you move words around, the meaning often stays similar, but the emphasis changes.

Both are possible, depending on context.

Pyytää is the present tense. Finnish present tense can correspond to:

• asks
• is asking

So this sentence could mean:

• The teacher asks us to answer the questions quickly.
• The teacher is asking us to answer the questions quickly.

Finnish does not usually make the same simple-present vs. present-continuous distinction that English does.

A very useful pattern is:

[subject] + pyytää + [person in partitive] + [verb in -maan/-mään]

Examples:

• Pyydän sinua odottamaan.
  = I ask you to wait.
• Hän pyytää minua auttamaan.
  = He/She asks me to help.
• Opettaja pyytää meitä lukemaan.
  = The teacher asks us to read.

So your sentence is a perfect model:

Opettaja pyytää meitä vastaamaan kysymyksiin nopeasti.

If you remember that pattern, you can build many similar sentences.