Bevor Besuch kommt, nehme ich das Bügeleisen aus dem Schrank und lege die Bluse auf den Tisch.

Because bevor is a subordinating conjunction. In German, a subordinate clause usually sends the conjugated verb to the end.

So:

• Bevor Besuch kommt
• literally: Before visitors come

This is the normal pattern:

• bevor
  • ... + verb at the end

More examples:

• Bevor ich gehe, räume ich auf.
• Bevor er anfängt, trinkt er Kaffee.

When a subordinate clause comes first, the main clause follows in normal German main-clause word order, where the conjugated verb must be in the second position.

So the first position is taken by the whole clause Bevor Besuch kommt, and then the verb comes next:

• Bevor Besuch kommt, nehme ich ...

Not:

• Bevor Besuch kommt, ich nehme ... ✘

This is a very common pattern in German:

• Wenn ich Zeit habe, gehe ich spazieren.
• Obwohl es regnet, fahren wir los.

In this sentence, Besuch is used in an idiomatic way to mean visitors or company.

• Besuch kommt = we are having visitors / company is coming

German often uses Besuch without an article in this expression when speaking generally.

If you say der Besuch, it sounds more specific, like a particular visit or the specific visitors you already have in mind.

So:

• Bevor Besuch kommt = before visitors come / before company arrives
• Bevor der Besuch kommt = before the visitors come / before the visit

Both are possible, but the version without the article is very natural in everyday speech.

They are related, but not identical.

• Besuch = the visit, or visitors/company as a general idea
• Besucher = visitors in a more literal countable sense
• Gäste = guests

In this sentence, Besuch kommt is especially idiomatic. It sounds natural for people are coming over.

Compare:

• Wir bekommen Besuch. = We are having company.
• Die Besucher kommen um 15 Uhr. = The visitors arrive at 3.
• Unsere Gäste kommen bald. = Our guests are coming soon.

This is about prepositions and case.

1. aus dem Schrank

The preposition aus always takes the dative.

• der Schrank → dative: dem Schrank

So:

• aus dem Schrank = out of the cupboard/closet

2. auf den Tisch

The preposition auf is a two-way preposition. It can take:

• accusative for movement toward a destination
• dative for location

Here there is movement: the blouse is being placed onto the table.

So:

• auf den Tisch = onto the table

Compare:

• Die Bluse liegt auf dem Tisch. = The blouse is on the table.
  (location, so dative)
• Ich lege die Bluse auf den Tisch. = I put the blouse onto the table.
  (movement, so accusative)

German often chooses different verbs depending on the position of the object.

• legen = to lay/put something down so that it ends up lying flat
• stellen = to stand/put something upright
• setzen = to set/place in a seated position

A blouse normally ends up lying flat on a table, so legen is the natural verb.

Examples:

• Ich lege das Buch auf den Tisch.
• Ich stelle die Flasche auf den Tisch.
• Ich setze das Kind auf den Stuhl.

So lege die Bluse auf den Tisch is exactly what a German speaker would expect.

Because the same subject applies to both verbs:

• nehme ich das Bügeleisen aus dem Schrank
• und lege die Bluse auf den Tisch

German often omits the repeated subject when two verbs are connected and the subject stays the same.

So this is completely normal:

• Ich koche und esse.
• Sie öffnet die Tür und geht hinein.

You could repeat ich, but it would usually sound unnecessary here.

Both are direct objects, so they are in the accusative case.

• das Bügeleisen is neuter
  Nominative: das Bügeleisen
  Accusative: das Bügeleisen
  (same form)

• die Bluse is feminine
  Nominative: die Bluse
  Accusative: die Bluse
  (same form)

So even though the forms do not change here, grammatically they are accusative objects because they are the things being taken and placed.

Yes, that is possible, but the nuance is a little different.

• Bevor Besuch kommt sounds general and idiomatic: before company comes
• Bevor der Besuch kommt sounds more specific: before the visitors come or before the visit

If you are talking about a known, specific group of people, der Besuch may fit well. If you are speaking more generally about having people over, no article is very natural.

Bevor normally introduces a clause:

• Bevor Besuch kommt ...
• Bevor ich esse ...

If you want to use just a noun phrase in German, vor is often used instead:

• vor dem Essen = before the meal
• vor dem Besuch = before the visit

So:

• bevor
  • clause
• vor
  • noun

That is a very useful distinction to remember.

Yes. In German, a subordinate clause is normally separated from the main clause by a comma.

So the comma in:

• Bevor Besuch kommt, nehme ich ...

is required, not optional.

This is especially common with conjunctions like:

• weil
• dass
• wenn
• obwohl
• bevor

For English speakers, this is worth paying attention to because German comma rules are stricter in these structures.