Η μπουγάδα δεν στεγνώνει γρήγορα όταν το σχοινί είναι μέσα στο σπίτι, ακόμα κι αν βάλω πολλά μανταλάκια.

Because μπουγάδα is a collective noun in Greek. It means the laundry / the wash / a load of washing as one whole thing, so it is grammatically singular.

That is why Greek says:

• Η μπουγάδα στεγνώνει = The laundry is drying / dries

Even though the laundry contains many items, the noun itself is singular feminine.

Στεγνώνει is the 3rd person singular present of στεγνώνω, meaning to dry or to become dry.

So δεν στεγνώνει γρήγορα means:

• it doesn’t dry quickly
• or more naturally in context, the laundry doesn’t dry quickly

The Greek present tense here expresses a general situation or usual result, not just something happening at this exact moment.

Δεν is the normal negation used with verbs in Modern Greek.

So:

• στεγνώνει = it dries / is drying
• δεν στεγνώνει = it doesn’t dry / isn’t drying

This is very common Greek word order: δεν + verb.

Here όταν means when, but in a sentence like this it often has the broader sense of whenever.

So:

• όταν το σχοινί είναι μέσα στο σπίτι
  = when / whenever the clothesline is inside the house

Because the whole sentence describes a general truth or repeated situation, English could understand it as whenever even if the Greek uses όταν.

Yes, literally το σχοινί means the rope. But in this context it refers to the clothesline.

Greek often uses the ordinary word rope/line for a washing line, especially when the context makes it obvious.

So here:

• το σχοινί είναι μέσα στο σπίτι
  = the clothesline is inside the house

Because στο is the contraction of σε + το:

• σε = in / at / to
• το = the
• στο = in the / at the / to the

So:

• μέσα στο σπίτι = inside the house

The phrase μέσα στο σπίτι is the natural full expression.
Literally, it is something like inside in-the house, but in real Greek it simply means inside the house.

Ακόμα κι αν means even if.

• αν = if
• ακόμα κι αν = even if

It introduces a concession: something is true despite another possible action or condition.

So:

• ακόμα κι αν βάλω πολλά μανταλάκια
  = even if I put a lot of clothespins

Also, κι here is just a shortened form of και. Before a vowel, Greek often shortens και to κι.

So:

• ακόμα και αν
• ακόμα κι αν

both mean the same thing.

This is a very important Greek pattern.

After αν in this kind of sentence, Greek very often uses the subjunctive, and here the form is βάλω, from βάζω.

• βάζω = I put / I am putting / I put regularly
• βάλω = I put in a single, complete action sense

So:

• ακόμα κι αν βάλω πολλά μανταλάκια
  means even if I put a lot of clothespins

Greek chooses βάλω because the speaker means the act of putting them on, not an ongoing or habitual process.

This is the aorist subjunctive form, which is extremely common after words like να and αν.

Because the speaker is now referring to their own action.

The sentence structure is:

• The laundry doesn’t dry quickly...
• ...even if I put lots of clothespins.

In Greek, the subject pronoun εγώ is usually omitted because it is already clear from the verb ending.

So:

• βάλω = I put
• full version would be ακόμα κι αν εγώ βάλω..., but that would usually sound unnecessary unless you wanted emphasis.

Μανταλάκια is the plural of μανταλάκι, which means clothespin / clothes peg / clip.

So:

• πολλά μανταλάκια = many clothespins / lots of pegs

A few useful details:

• μανταλάκι is neuter singular
• μανταλάκια is neuter plural
• πολλά is also neuter plural, so it agrees with μανταλάκια

So the grammar matches perfectly:

• πολλά
  • μανταλάκια

Yes: the word order is very natural for Greek information flow.

The sentence goes:

• main statement first: Η μπουγάδα δεν στεγνώνει γρήγορα
• then the condition/situation: όταν το σχοινί είναι μέσα στο σπίτι
• then the concessive idea: ακόμα κι αν βάλω πολλά μανταλάκια

This gives the listener the main point first, then the circumstances, then the even if part.

Greek word order is flexible, but this version sounds natural and clear.