Я выбросил пакетик после того, как чай немного остыл.

Выбросил means threw away / discarded, not just threw.

• бросить = to throw
• выбросить = to throw out, throw away

The prefix вы- often adds the idea of out. So in this sentence, the person is not simply throwing the tea bag somewhere; they are getting rid of it.

Also, выбросил is perfective, which fits well because the action is seen as a completed whole: the tea bag was thrown away.

Yes. In this context, пакетик means tea bag.

Literally, пакетик is a diminutive form of пакет, so it means something like little packet or small sachet. In everyday Russian, people often say simply пакетик when it is obvious from context that they mean a tea bag.

You could also say:

• чайный пакетик = tea bag

But in a sentence about tea, пакетик alone is very natural.

Пакетик is in the accusative singular, because it is the direct object of выбросил.

The speaker threw away what?
→ пакетик

For inanimate masculine nouns, the accusative singular looks the same as the nominative singular.

So:

• nominative: пакетик
• accusative: пакетик

That is why the form does not change.

Because после того, как is the normal way to say after before a full clause.

Russian uses:

• после
  • noun phrase in the genitive
    
    • после чая = after the tea
• после того, как
  • clause
    
    • после того, как чай немного остыл = after the tea cooled a bit

So после как is not standard here.

The word того is the genitive form of то, and together после того, как functions as a fixed conjunction meaning after.

Yes, you could say:

Я выбросил пакетик, когда чай немного остыл.

That would still be understandable, but the nuance is a little different.

• когда = when
• после того, как = after

So после того, как more clearly emphasizes sequence: first the tea cooled a little, then the speaker threw away the tea bag.

With когда, the connection is more neutral and can sometimes sound a bit looser.

Because остыл is perfective, and the sentence refers to a completed change of state.

• остывать = to be cooling down, to cool down gradually, process
• остыть = to cool down, become cool, reach that result

In this sentence, the tea cooled a bit, and after that the tea bag was thrown away. That completed result fits остыл very well.

If you used остывал, it would sound more like background description of an ongoing process, not a completed point in time.

Немного means a little, a bit, or slightly.

So чай немного остыл means:

• the tea cooled down a little
• the tea became slightly less hot

It shows that the tea did not become completely cold; it just cooled enough for the speaker to act.

A close, slightly more conversational alternative is немножко.

That is a normal past-tense marker in Russian.

In the past tense, Russian verbs also show gender in the singular:

• masculine: -л
• feminine: -ла
• neuter: -ло
• plural: -ли

So here:

• я выбросил = I threw away, said by a male speaker
• чай остыл = the tea cooled down
  Since чай is masculine, the verb is остыл

If the speaker were female, it would be:

• Я выбросила пакетик...

But остыл would stay the same, because чай is still masculine.

In many Russian sentences, subject pronouns can be omitted if the meaning is clear from context. But in the past tense, that can be less obvious, because past-tense verbs do not show person, only gender and number.

So:

• Я выбросил пакетик... = clearly I threw away the tea bag
• Выбросил пакетик... = could mean I threw it away or he threw it away, depending on context

That is why я is useful here. It makes the subject explicit.

Yes. Russian word order is fairly flexible.

A very natural alternative is:

После того, как чай немного остыл, я выбросил пакетик.

This means the same thing. The difference is mostly about focus and style:

• Я выбросил пакетик после того, как...
  starts with the main action
• После того, как..., я выбросил пакетик
  starts with the time/background information

Both are correct.

Because как helps introduce the subordinate clause.

In this expression:

• после того sets up the idea of after that
• как introduces the actual event that happened

So the structure is basically:

• after that, that...
• more naturally in English: after

You do not need to translate each word separately every time. It is best to learn после того, как as one whole connector meaning after when followed by a clause.

Usually it means the tea cooled down a bit, not necessarily that it became fully cool.

That is because of немного:

• остыл by itself can mean cooled down
• немного остыл specifically means cooled a little

So in context, the idea is probably that the tea was no longer extremely hot, and then the speaker removed or threw away the tea bag.