Learn Hardcore German: Stell den Eimer bitte neben die Garderobe; der Lappen liegt noch im Bad. - Please place the bucket next to the coat rack; the cloth is still lying in the bathroom.

Stell is the imperative of stellen for du (informal singular). German imperatives are verb-first and the subject du is usually omitted. Variants:

Stell(e) den Eimer … (du; the -e in stelle is optional)
Stellt den Eimer … (ihr, plural informal)
Stellen Sie den Eimer … (polite/formal)

Why is it den Eimer but die Garderobe?

den Eimer is masculine accusative (direct object of stellen).
neben die Garderobe is accusative because neben is a two-way preposition; with movement toward a place (caused by stellen), it takes the accusative. If it were location only, it would be dative: neben der Garderobe.

Why accusative after neben here, but sometimes dative?

Wechselpräpositionen (two-way prepositions) like neben use:

Accusative for direction/change of location: Stell den Eimer neben die Garderobe.
Dative for location/rest: Der Eimer steht neben der Garderobe.

Could I use an or bei instead of neben?

neben = next to, side by side (precise).
an = at/onto a vertical surface or attached to it; with a bucket this would sound odd unless you literally attach it.
bei = at/near, less specific and uses dative: bei der Garderobe. It’s vaguer than neben.

Why stellen for the bucket but liegen for the rag?

German distinguishes how objects are placed/are positioned:

stellen (upright/standing) → a bucket typically stands: den Eimer stellen; later: Der Eimer steht …
legen (lay, flat) → rags/books usually lie: den Lappen legen; later: Der Lappen liegt …
setzen (set, for people/animals sitting). So Der Lappen liegt … is natural; Der Lappen ist … is possible but less descriptive.

What does noch mean here, and where does it go?

Here noch means “still.” It’s a temporal adverb and typically precedes the place phrase:

Der Lappen liegt noch im Bad. You can move it for emphasis: Im Bad liegt der Lappen noch.

Why im Bad and not ins Bad?

im = in dem (dative, location): Der Lappen liegt im Bad.
ins = in das (accusative, direction): Ich gehe ins Bad. Here we’re describing location (no movement), so dative.

What exactly does Garderobe mean in a home context?

In homes, die Garderobe usually means the coat rack/cloak area in the hallway/entryway. It is not a bedroom wardrobe/closet; that’s typically der Kleiderschrank or der Schrank.

Can I move bitte around?

Yes. Common positions include:

Bitte stell den Eimer neben die Garderobe.
Stell bitte den Eimer neben die Garderobe.
Stell den Eimer bitte neben die Garderobe. No commas are needed around bitte. You may also soften with colloquial mal: Stell den Eimer mal …

Is the semicolon between the clauses okay?

Yes. A semicolon can separate two closely related main clauses. A period would be most common in everyday writing. In German, a comma between two main clauses without a conjunction is also acceptable in asyndetic coordination, especially when they’re short and closely linked: Stell den Eimer …, der Lappen liegt noch im Bad.

How would I make the request more polite?

Use the formal Sie or a question:

Stellen Sie den Eimer bitte neben die Garderobe.
Könntest du den Eimer bitte neben die Garderobe stellen? (informal, softer)
Könnten Sie den Eimer bitte neben die Garderobe stellen? (formal, very polite)

What are the genders and plurals of the nouns here?

der Eimer (pl: die Eimer)
der Lappen (pl: die Lappen)
die Garderobe (pl: die Garderoben)
das Bad (pl: die Bäder)

Are there regional or common synonyms I should know?

Eimer ≈ Kübel (Austrian/southern).
Lappen can be specified as Putzlappen/Wischlappen (cleaning rag).
Bad ≈ Badezimmer (neutral), Toilette/WC if you specifically mean the toilet room. Note: Lappen can be slang/insult for a person; here it’s just a cleaning rag.

Can I drop the article and say Stell Eimer …?

No. German usually requires an article with countable nouns. Use den Eimer (definite, a known bucket) or einen Eimer (indefinite, some bucket), depending on context.