Fass die Türklinke bitte vorsichtig an; du musst die Tür nicht ganz öffnen.

Because anfassen is a separable verb. Its basic meaning here is to touch or to handle with your hand.

• Infinitive: anfassen
• In a main clause or command, the prefix an- separates and goes to the end:
  • Du fasst die Türklinke an.
  • Fass die Türklinke an!

This is very common in German with verbs like aufmachen, einkaufen, mitkommen, etc.

Fass is the informal singular imperative of anfassen, used when speaking to one person with du.

The full pattern is:

• infinitive: anfassen
• du form: du fasst an
• imperative for du: fass ... an

So:

• Fass die Türklinke an. = speaking to one person informally
• Fasst die Türklinke an. = speaking to several people informally (ihr)
• Fassen Sie die Türklinke an. = formal

German very often makes compound nouns by joining words together.

• Tür = door
• Klinke = handle / latch / lever handle
• Türklinke = door handle / door lever

This is completely normal in German. English often writes similar ideas as two words, but German usually combines them into one.

Also note that the last part of the compound determines the gender, so because Klinke is feminine, Türklinke is also feminine: die Türklinke.

Both nouns are feminine:

• die Türklinke
• die Tür

In this sentence, both are also direct objects:

• Fass die Türklinke an
• die Tür nicht ganz öffnen

For feminine singular nouns, the article die stays the same in both nominative and accusative, so you do not see a change here.

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

• Fass die Türklinke an. = direct
• Fass die Türklinke bitte an. = please touch the door handle

Its position is fairly flexible. You could also hear:

• Bitte fass die Türklinke vorsichtig an.
• Fass bitte die Türklinke vorsichtig an.

The version in your sentence sounds natural and polite.

Here vorsichtig is being used as an adverb, not an adjective before a noun.

It describes how you should touch the handle:

• vorsichtig anfassen = to touch carefully / cautiously

Adverbs in German do not take adjective endings.

Compare:

• die vorsichtige Person = the careful person
  Here vorsichtige is an adjective before a noun, so it gets an ending.
• Fass die Türklinke vorsichtig an.
  Here vorsichtig describes the action, so it stays unchanged.

Both can relate to touching, but they are not always identical.

• anfassen often means to touch with the hand, to handle, or sometimes to grab
• berühren is often more neutral or more formal, like to touch/contact

With a door handle, anfassen sounds very natural, because you are physically putting your hand on it and handling it.

Because the clause uses the modal verb müssen.

In German main clauses with a modal verb:

• the modal verb is conjugated in second position
• the other verb stays in the infinitive at the end

So:

• du musst ... öffnen

That gives:

• Du musst die Tür nicht ganz öffnen.

This is a basic German word order pattern:

• Ich kann Deutsch sprechen.
• Wir wollen nach Hause gehen.
• Du musst die Tür nicht ganz öffnen.

Not exactly. Here it means:

• you do not have to open the door all the way
• you don't need to open it completely

This is an important point:

• nicht müssen = not have to
• not usually must not

So the sentence is saying opening the door fully is unnecessary, not forbidden.

If you wanted a real prohibition, German would more naturally use something like:

• Du darfst die Tür nicht ganz öffnen. = You must not / may not open the door all the way.

Here ganz means completely, fully, or all the way.

So:

• die Tür ganz öffnen = to open the door completely
• die Tür nicht ganz öffnen = not open the door completely / not open it all the way

It does not mean whole here in the sense of the whole door. It modifies the action öffnen.

Yes. Du is the informal form, used with friends, family, children, or people you are on informal terms with.

A formal version would be:

• Fassen Sie die Türklinke bitte vorsichtig an; Sie müssen die Tür nicht ganz öffnen.

So the original sentence sounds like someone speaking informally to one person.

The semicolon links two closely related independent clauses:

• Fass die Türklinke bitte vorsichtig an
• du musst die Tür nicht ganz öffnen

It is stronger than a comma and weaker than a full stop. In English and German, it can show that the two ideas belong closely together.

You could also write this as two sentences:

• Fass die Türklinke bitte vorsichtig an. Du musst die Tür nicht ganz öffnen.

The meaning stays the same.