Learn Hardcore German: Beim Putzen höre ich Musik im Radio. - While cleaning I listen to music on the radio.

So beim Putzen literally means “during the cleaning” / “while (I am) cleaning”.
You cannot say bei Putzen; the preposition bei needs a dative noun with an article, and that’s what is hidden inside beim.

Why is Putzen capitalized, and what kind of form is that?

Putzen here is a verb infinitive used as a noun. In German, infinitives can be “nominalized” (turned into nouns), and then they:

Are capitalized: Putzen
Take neuter gender: das Putzen
Can follow prepositions, just like normal nouns: beim Putzen = bei dem Putzen

So beim Putzen literally is “during the cleaning”, but structurally it’s bei + dem + (noun) Putzen. This is a very common pattern:

beim Essen – while eating
beim Lesen – while reading
beim Arbeiten – while working

Could I also say Wenn ich putze, höre ich Musik im Radio? Is it the same meaning?

Yes, Wenn ich putze, höre ich Musik im Radio is perfectly correct and very natural.

The differences are mostly style/form:

Beim Putzen höre ich Musik im Radio.
- Uses a prepositional phrase with a nominalized verb.
- Very compact, slightly more “written” or neutral.
Wenn ich putze, höre ich Musik im Radio.
- Uses a full clause with a finite verb (ich putze).
- Feels a bit more explicit and conversational.

In everyday German you’ll hear both; they mean essentially the same: “When / while I (am) cleaning, I listen to music on the radio.”

Why is the verb höre in second position, after Beim Putzen?

German main clauses follow the verb-second rule (V2):

Exactly one element (subject, time phrase, prepositional phrase, etc.) comes first.
The conjugated verb comes second.
Everything else follows.

In the sentence:

First element: Beim Putzen (counts as one unit)
Second position: höre (the finite verb)
Rest: ich Musik im Radio

So:

Beim Putzen | höre | ich Musik im Radio.

You cannot say Beim Putzen ich höre Musik im Radio; that would violate the V2 rule.

What case is used after beim Putzen and im Radio, and how can I see that?

Both beim and im contain a preposition + dative article:

beim = bei
- dem (dative singular)
- bei always takes the dative:
  - bei dem Putzen → beim Putzen
im = in
- dem (dative singular)
- in takes dative for a location (where?), and accusative for a direction (where to?):
  - Location: in dem Radio → im Radio (dative)
  - Direction: ins Radio (in das Radio) would be accusative.

So in the sentence, both Putzen and Radio are in the dative case, because of bei and in with no movement.

Why is it im Radio (“in the radio”) when English says “on the radio”?

This is just a difference in idiom between English and German.

English: on the radio
German: im Radio (literally “in the radio”)

In German, many media are thought of as something “in” which content appears:

im Radio – on the radio
im Fernsehen – on TV
im Internet – on the internet

So Musik im Radio is the normal way to say “music on the radio” in German.

Why is there no article before Musik (why not die Musik)?

In German, mass nouns / uncountable nouns often appear without an article when they are meant in a general, non-specific sense. Musik is one of these:

Ich höre Musik. – I listen to music (in general).
Ich trinke Kaffee. – I drink coffee (in general).
Ich mag Fleisch. – I like meat.

If you use die Musik, you usually refer to specific or previously mentioned music:

Ich höre die Musik, die du mir empfohlen hast.
I’m listening to the music you recommended.

In Beim Putzen höre ich Musik im Radio, the speaker means music in general, so no article is used.

Is the word order Musik im Radio fixed, or can I also say im Radio Musik?

Both word orders are grammatically possible:

Beim Putzen höre ich Musik im Radio.
Beim Putzen höre ich im Radio Musik.

The most natural in this context is Musik im Radio, because:

Musik im Radio is felt as a single unit (“music on the radio”).
Direct objects (like Musik) often come before adverbials/prepositional phrases (like im Radio).

Putting im Radio first can be used for emphasis on the medium:

Beim Putzen höre ich im Radio Musik, nicht auf Spotify.
(emphasizing: specifically on the radio, not on another platform)

Can I also say Ich höre Radio or Ich höre das Radio instead of Ich höre Musik im Radio?

Yes, but the nuances differ:

Ich höre Radio.
- Very common and idiomatic.
- Means “I listen to the radio (broadcasts).”
- Focus on the medium, not on what exactly you’re hearing.
Ich höre das Radio.
- Grammatically correct, but much less common.
- Sounds more like “I hear the radio (device)”, or can sound slightly odd in many contexts.
Ich höre Musik im Radio.
- Emphasizes that what you’re listening to is music, and that it’s coming from the radio.

So if you want to stay close to the English “I listen to music on the radio”, Ich höre Musik im Radio is a very natural choice.

Could I replace beim Putzen with während des Putzens or während ich putze? Is there a difference?

You can, and all are correct, but they differ slightly in style:

Beim Putzen höre ich Musik im Radio.
- Neutral, very common, and compact.
Während des Putzens höre ich Musik im Radio.
- More formal and a bit stiff in everyday speech.
- während
  - genitive (des Putzens).
Während ich putze, höre ich Musik im Radio.
- Very natural and common.
- Uses a full clause, similar to English “While I’m cleaning, …”.

In most everyday situations, beim Putzen and während ich putze are the most typical choices.