Ferme doucement la porte.

Because it’s the imperative. In French, commands normally omit the subject pronoun:

• Ferme = the imperative for tu (informal singular)
• Fermez = the imperative for vous (formal singular or plural)
• Fermons = the imperative for nous (“let’s close”)

For most -er verbs, the imperative with tu drops the final s:

• Present: tu fermes
• Imperative: Ferme ! Exception: you add back an s before y or en for ease of pronunciation, e.g. Vas‑y !, Parles‑en ! (not relevant here because there’s no y/en).

Yes. French often uses the infinitive for impersonal instructions:

• Fermer la porte doucement. = “Close the door gently” (as a general instruction on a notice or in a manual) It feels less like you’re addressing someone directly than the imperative does.

Doucement is an adverb meaning “gently/softly/quietly,” and depending on context it can also imply “slowly.” With a door, it usually means “gently/quietly (so it doesn’t make noise).”

• If you specifically mean slow speed: lentement
• Other near‑synonyms by nuance: calmement, sans bruit, en douceur

Both are natural:

• Ferme doucement la porte.
• Ferme la porte doucement. Style preference varies; meaning is the same. For emphasis, you could front the adverb with a comma: Doucement, ferme la porte.

In the affirmative imperative, object pronouns follow the verb with hyphens:

• Ferme‑la doucement. = “Close it gently.” (referring to a feminine noun like la porte) In the negative imperative, pronouns go before the verb:
• Ne la ferme pas (doucement).

It’s common with imperatives:

• Ferme doucement la porte ! But it’s not mandatory in everyday writing; without it, it just looks a bit calmer.

Yes:

• Informal: Ferme doucement la porte, s’il te plaît.
• Formal/plural: Fermez doucement la porte, s’il vous plaît.