Bitte drück die Türklinke langsam herunter.

drück is the imperative form, used to give a command to du.

The verb is drücken = to press.
For the du command, German usually drops the -st ending, so:

• du drückst = you press
• drück! = press!

You may also see drücke! in more careful or formal written German, but drück! is very common in everyday speech.

So:

• Bitte drück ... = Please press ...

Because the full verb is herunterdrücken, which is a separable verb.

In many sentence types, separable verbs split into:

• the main verb part in the normal verb position
• the prefix at the end

So:

• infinitive: herunterdrücken
• command: Drück ... herunter!

This is very common in German:

• aufmachen → Mach die Tür auf!
• anmachen → Mach das Licht an!
• herunterdrücken → Drück die Türklinke herunter!

Literally, herunterdrücken means to press down.

It is made up of:

• herunter = down
• drücken = to press

So in this sentence, it means something like:

• press the door handle down slowly

It sounds more specific than just drücken, because it tells you the direction: downward.

Türklinke is a feminine noun in German, so its dictionary form is:

• die Türklinke = the door handle

In the sentence, it is the direct object of the verb, so it is in the accusative case.
For feminine nouns, the article stays the same in nominative and accusative:

• nominative: die Türklinke
• accusative: die Türklinke

So even though it is accusative here, it still appears as die.

Because German often forms compound nouns by joining words together.

Here:

• Tür = door
• Klinke = handle / latch handle

Together:

• die Türklinke = door handle

This is completely normal in German. English usually writes such things as separate words, but German often combines them into one noun.

Langsam is an adverb meaning slowly.

German adverb placement is flexible, but here langsam is placed before the separated prefix herunter, which sounds natural:

• Bitte drück die Türklinke langsam herunter.

This is similar to saying:

• Please press the door handle down slowly.

You might also hear slightly different placements in other contexts, but this one is very normal.

Bitte makes the command more polite.

Compare:

• Drück die Türklinke herunter. = a plain command
• Bitte drück die Türklinke herunter. = Please press the door handle down

So bitte softens the instruction and makes it sound less abrupt.

It is addressed to du.

You can tell from the imperative form drück.

Compare:

• du: Drück die Türklinke herunter.
• ihr: Drückt die Türklinke herunter.
• Sie: Drücken Sie die Türklinke herunter.

So this sentence is speaking to one person informally.

Yes. runterdrücken is a very common colloquial alternative.

So you may hear:

• Bitte drück die Türklinke langsam runter.

That is very natural in spoken German.

In general:

• herunter sounds a bit more neutral or careful
• runter sounds more conversational

Both are common and understood.

Because all nouns are capitalized in German.

So:

• die Tür
• die Klinke
• die Türklinke

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

Yes, that is also correct.

German allows some flexibility with bitte and with the slightly fuller imperative drücke.

These are all possible:

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

They all mean essentially the same thing.
The version with drück is especially natural in everyday spoken German.