Learn Hardcore German: Ich fülle einen Eimer mit warmem Wasser und hole einen Lappen. - I fill a bucket with warm water and fetch a cloth.

Because füllen takes a direct object (what you fill), which is in the accusative case. Eimer is masculine, and the accusative of the indefinite article is einen: ich fülle einen Eimer. (Nominative would be ein Eimer; dative would be einem Eimer.)

Why is it mit warmem Wasser and not mit warmen Wasser?

mit always takes the dative case.
Wasser is neuter singular.
With no article before the adjective, you use the strong adjective ending in the dative neuter: -em. So: mit warmem Wasser.
If you use an article/determiner, the ending changes: mit dem warmen Wasser, mit meinem warmen Wasser.

Note: mit warmen Wasser (without an article) is a common mistake in standard German.

Why is there no article before Wasser?

Wasser is an uncountable mass noun. In a generic sense German typically omits an article: mit warmem Wasser. You’d add an article if the water is specific: mit dem warmen Wasser (aus dem Hahn rechts), or use a measure: mit einem Eimer Wasser, mit einem Glas Wasser.

Can I also say Ich fülle warmes Wasser in einen Eimer? Is there a difference?

Yes, that’s correct too. Two common patterns:

Container as object + content with mit: Ich fülle einen Eimer mit warmem Wasser.
Content as object + container with in (+ accusative): Ich fülle warmes Wasser in einen Eimer.

They mean the same thing; the first highlights the container, the second highlights the content. Don’t say “Ich fülle einen Eimer in warmes Wasser” (that sounds like you’re putting the bucket into water).

Why holen here? How is it different from nehmen, bringen, besorgen, or abholen?

holen = go get/fetch (usually implies moving away and back): ich gehe hin und hole es.
nehmen = take (no fetching implied): ich nehme einen Lappen (that’s right here).
bringen = bring (move something to someone): ich bringe dir einen Lappen.
besorgen = procure/obtain (often involves effort or arranging): ich besorge einen Lappen.
abholen = pick up (at an agreed place/time): ich hole das Paket ab.

In the sentence, holen fits “go and fetch a rag.”

Should it be Ich hole mir einen Lappen?

You can add the so‑called “dative of interest”:

Ich hole mir einen Lappen. = I’ll get myself a rag (for my own use/benefit).
Ich hole einen Lappen. = I’ll get a rag (neutral about for whom).

Both are correct; with household tasks, many speakers naturally say mir.

What exactly is a Lappen? Any nuances or safer alternatives?

der Lappen is a cleaning rag/cloth. More specific or neutral options:

der Putzlappen (cleaning rag)
das Tuch (cloth; broader)
der Waschlappen (washcloth; also a mild insult in slang) Note: Lappen can be slangy/derogatory in other contexts (an insult for a person). Here, it’s the literal cleaning rag.

Can I move mit warmem Wasser earlier in the sentence?

Yes, word order is flexible for emphasis:

Neutral: Ich fülle einen Eimer mit warmem Wasser.
Focus on the means: Ich fülle mit warmem Wasser einen Eimer.
Strong fronting: Mit warmem Wasser fülle ich einen Eimer. All are grammatical; the first is the most idiomatic here.

Do I need a comma before und?

No. Here the two parts share the same subject and the second clause omits it: Ich fülle … und hole …—no comma. If you repeat the subject, a comma becomes optional: Ich fülle …, und ich hole … (comma mainly for clarity or rhythm).

Does und imply that I first fill the bucket and then fetch the rag?

Usually yes, readers will assume a natural sequence. If you want to make the order explicit, use sequencing words:

Zuerst fülle ich …, dann hole ich … If you mean simultaneity, rephrase (e.g., with während) because plain und often feels sequential.

Is the present tense here “I fill” or “I am filling”? Can it also be future?

German has no separate progressive form, so ich fülle can mean both “I fill” and “I am filling,” depending on context. The present can also refer to near future plans: Heute Abend fülle ich einen Eimer …

Why fülle with ü? How do I type and pronounce it?

Spelling: füllen has ü. If you can’t type it, write fuellen (not “fullen”).
Pronunciation: fü is like French “u” (rounded front vowel). Roughly “FY-luh” for fülle. holen has a long o: “HO-len.”
Don’t confuse füllen (to fill) with fühlen (to feel).

Could I say den Eimer instead of einen Eimer?

Yes, if you and your listener know which specific bucket you mean: Ich fülle den Eimer mit warmem Wasser … Indefinite einen just presents it as “a bucket,” not a particular one.

I thought dative adjectives often end in -en. Why is it warmem here?

That -en ending appears in dative when there’s a determiner (article, possessive, demonstrative). Without a determiner, you use strong endings:

No article (strong): mit warmem Wasser (dative neuter = -em)
With article/possessive (mixed/weak): mit dem warmen Wasser, mit meinem warmen Wasser

So the ending changes based on whether a determiner stands in front.