Drück bitte die Türklinke herunter, dann kannst du die Tür öffnen.

Drück is the informal singular command form, the du imperative, of drücken.

From du drückst, the imperative is drück(e)!
So both of these are correct:

• Drück bitte ...
• Drücke bitte ...

The version without -e is very common in everyday German and sounds natural.

Because herunterdrücken is a separable verb.

Its full dictionary form is herunterdrücken = to press down.
In a main clause or command, the conjugated verb part goes near the front, and the separable prefix goes to the end:

• infinitive: die Türklinke herunterdrücken
• command: Drück die Türklinke herunter

This is a very common German pattern.

Herunter adds the idea of downward movement.

So:

• drücken = to press / push
• herunterdrücken = to press down

In this sentence, it tells you not just to touch the handle, but to push it downward.

Türklinke is a feminine noun, so its article is die.

It is also a compound noun:

• Tür = door
• Klinke = handle / latch handle

In German compounds, the last part decides the gender.
Since die Klinke is feminine, die Türklinke is feminine too.

Also, here it is the direct object of drücken, so it is in the accusative case. But for feminine singular, the accusative article is still die, so the form does not change.

Because all nouns are capitalized in German.

So:

• die Tür
• die Türklinke

This is one of the most noticeable spelling differences between German and English.

Bitte makes the command more polite, like please in English.

Its position is fairly flexible. All of these are possible:

• Drück bitte die Türklinke herunter.
• Bitte drück die Türklinke herunter.
• Drück die Türklinke bitte herunter.

The meaning stays basically the same, though the emphasis and rhythm change a little.

Because the sentence contains two main clauses:

• Drück bitte die Türklinke herunter
• dann kannst du die Tür öffnen

They are separate clauses, so German uses a comma here.
Also, dann is an adverb, not a conjunction like und.

German main clauses follow the verb-second rule.

That means the finite verb must come in the second position.
Here, dann takes the first position, so kannst has to come next:

• Dann kannst du die Tür öffnen.

So the order is:

1. dann
2. kannst
3. du

This is very normal German word order.

Because kannst is a form of the modal verb können.

With modal verbs in German, the second verb usually goes to the end in the infinitive:

• du kannst die Tür öffnen
• ich will nach Hause gehen
• sie muss arbeiten

So öffnen stays at the end because it depends on kannst.

Yes, du is the informal singular you.

A formal version would be:

Drücken Sie bitte die Türklinke herunter, dann können Sie die Tür öffnen.

Changes:

• Drück → Drücken Sie
• kannst du → können Sie

This is what you would say to a stranger, customer, or someone in a formal situation.

Yes. Runter is very common in everyday spoken German and usually means the same thing here.

So these are both natural:

• Drück die Türklinke herunter.
• Drück die Türklinke runter.

Herunter sounds a bit more neutral or standard; runter sounds more colloquial.

English does not have an exact equivalent, but a good trick is this:

1. Say ee as in see.
2. Keep your tongue there.
3. Round your lips as if you were saying oo.

That gets you close to German ü.

So:

• drück has that rounded ü sound
• Tür does too

It should not sound like English oo in food, and not like the vowel in turn either.