Po herbacie z mlekiem poczuję się lepiej.

Because po meaning after normally requires the locative case in Polish.

• herbata = tea
• po herbacie = after tea

So herbacie is the locative singular form of herbata.

Here po means after in a time-related sense: after having tea / once I’ve had tea.

So Po herbacie z mlekiem means something like:

• After tea with milk
• After having tea with milk
• Once I’ve had tea with milk

It does not mean because of tea with milk. The main idea is timing: first the tea, then feeling better.

Because the preposition z meaning with takes the instrumental case.

• mleko = milk
• z mlekiem = with milk

So:

• herbata z mlekiem = tea with milk

This is a very common pattern in Polish:

• kawa z cukrem = coffee with sugar
• kanapka z serem = sandwich with cheese
• ryż z warzywami = rice with vegetables

Usually it means tea with milk.

That is the most natural reading:

• herbata z mlekiem = tea with milk

In English, milk tea can sometimes refer to specific drink styles, but in Polish this phrase is normally just the straightforward description: regular tea with milk added.

Poczuję is the 1st person singular future form of poczuć, which is a perfective verb.

So it means:

• I will feel
• I will start to feel
• I will come to feel

Because poczuć is perfective, Polish uses a simple future form:

• poczuję = I will feel

This is different from an imperfective verb, which would usually make the future with będę.

Because czuć się / poczuć się is the normal Polish expression for to feel in the sense of one’s condition or state.

Compare:

• czuję zimno = I feel coldness / I feel the cold
• czuję się źle = I feel bad
• poczuję się lepiej = I will feel better

So się is part of the verb expression here. You generally need it when talking about how someone feels physically or emotionally.

Because lepiej is an adverb, while lepszy is an adjective.

After czuć się / poczuć się, Polish uses an adverb:

• czuję się dobrze = I feel well
• czuję się źle = I feel bad
• czuję się lepiej = I feel better

Here lepiej is the comparative form of dobrze:

• dobrze = well
• lepiej = better

You would use lepszy to describe a noun:

• lepsza herbata = better tea

Yes. That is also correct, but the nuance is a little different.

• Poczuję się lepiej = I will feel better / I will start to feel better
  
  • focuses on the change or the moment the improvement happens
• Będę się czuć lepiej = I will be feeling better / I will feel better
  
  • focuses more on the future state itself

In many everyday situations, both work, but poczuję się lepiej often sounds a bit more natural when you mean I’ll feel better after that.

Because Polish often leaves out subject pronouns when they are clear from the verb ending.

• poczuję already means I will feel
• so ja is not necessary

You could say Ja poczuję się lepiej, but that would usually add emphasis, contrast, or emotion.

For example:

• Ja poczuję się lepiej, ale on nie.
  = I will feel better, but he won’t.

No, Polish word order is fairly flexible.

Po herbacie z mlekiem poczuję się lepiej is natural and emphasizes the after tea with milk part first.

You could also say:

• Poczuję się lepiej po herbacie z mlekiem.

Both are grammatical. The difference is mainly in focus:

• Po herbacie z mlekiem... = sets the scene first
• Poczuję się lepiej... = puts the main statement first

Usually yes, in normal context.

Polish often uses po + noun to mean after having / after consuming / after experiencing something.

So here it naturally suggests:

• after drinking tea with milk

It does not usually mean merely after the tea exists or after seeing the tea. Context makes it clear that the tea is something you have had.

The full base form is poczuć się lepiej.

Its parts are:

• poczuć = to feel, to sense; in this context to come to feel
• się = reflexive particle
• lepiej = better

So if you were learning it as a phrase, it is useful to remember:

• czuć się dobrze / źle / lepiej
• poczuć się lepiej

rather than memorizing poczuć alone.

Yes, it is natural and grammatical.

It sounds like something someone might say when talking about feeling unwell and expecting improvement after having tea with milk.

Very natural alternatives would also be:

• Po herbacie z mlekiem będę się czuć lepiej.
• Poczuję się lepiej po herbacie z mlekiem.

But the original sentence is perfectly good Polish.