Lefekvés előtt a papucsot az ágy mellé teszem.

Because Hungarian often uses postpositions rather than prepositions.

In English, you say before going to bed, with before in front.
In Hungarian, előtt comes after the word it belongs to:

• lefekvés előtt = before going to bed
• literally: going-to-bed before

So this is normal Hungarian structure, not an exception.

Lefekvés is a verbal noun formed from the verb lefeküdni / lefeküdik, meaning to lie down / to go to bed.

The ending -ás / -és often turns a verb into a noun-like form:

• olvas → olvasás = reading
• fizet → fizetés = payment / paying
• lefekszik → lefekvés = going to bed

So lefekvés előtt is literally something like before the act of going to bed.

Because papucsot is in the accusative case, which marks the direct object.

Here, the thing being put somewhere is the slipper(s), so Hungarian marks it with -t:

• papucs = slipper
• papucsot = slipper + accusative

The -o- is just a linking vowel that makes the word easier to pronounce.
So the important part is really the final -t, which shows this is the object.

Because Hungarian uses two kinds of verb conjugation: indefinite and definite.

Here the object is a papucsot = the slipper(s), which is definite, so the verb also takes the definite conjugation:

• teszek = I put / I place something (indefinite)
• teszem = I put / I place it / the specific thing (definite)

So:

• Papucsot teszek... = I put a slipper / slippers...
• A papucsot teszem... = I put the slipper / slippers...

That is why teszem is the correct form here.

Teszem is:

• present tense
• first person singular
• definite conjugation

So it means I put or I am putting, depending on context.

The base verb is tesz = to put / place.

This is a very common and important distinction in Hungarian.

• mellé = to beside / next to → direction, movement toward a place
• mellett = beside / next to → location, already there

Since the sentence describes putting something somewhere, there is movement toward that position, so Hungarian uses mellé:

• az ágy mellé teszem = I put it next to the bed
• az ágy mellett van = it is next to the bed

So mellé is the natural choice because the action causes the slipper(s) to end up there.

Both a and az mean the. Hungarian has two forms of the definite article depending on the following sound:

• a before a consonant
• az before a vowel

So:

• a papucsot because papucs starts with p
• az ágy because ágy starts with á

This is purely about pronunciation.

Hungarian word order is flexible, and the beginning of the sentence often sets the topic or frame.

Starting with Lefekvés előtt gives the time context first:

• Lefekvés előtt a papucsot az ágy mellé teszem.

This feels natural, like saying:

• Before going to bed, I put the slippers next to the bed.

You could rearrange the sentence, but the emphasis would shift. For example:

• A papucsot lefekvés előtt az ágy mellé teszem.

This puts more attention on the slipper(s).

So the original order is a normal, natural way to present the action in its time frame.

Yes, in many contexts you could say rakom, and it would still sound natural.

• tesz = a very general put/place
• rak = also put/place, often a bit more physical or colloquial in some contexts

So:

• a papucsot az ágy mellé teszem
• a papucsot az ágy mellé rakom

Both can work.
However, teszem is a very neutral, standard choice.

Because Hungarian does not always state possession if it is already obvious from context.

• papucsot = the slipper(s)
• papucsomat = my slipper(s)

If it is clear that you mean your own slippers, Hungarian can simply say a papucsot. This is very common. The sentence focuses on the action and location, not on ownership.

If you wanted to stress that they are specifically my slippers, then papucsomat would be possible.