Smoczek leży obok poduszki, a grzechotka spadła pod łóżko.

Because obok takes the genitive case when it means next to / beside.

• dictionary form: poduszka
• genitive singular: poduszki

So:

• obok poduszki = next to the pillow

This is a very common pattern in Polish:

• obok domu = next to the house
• obok stołu = next to the table

Because Polish uses different cases after pod depending on whether you mean:

• movement toward a place → accusative
• location in a place → instrumental

Here, the rattle fell under the bed, so there is movement, which is why Polish uses:

• pod łóżko

Compare:

• Grzechotka spadła pod łóżko. = The rattle fell under the bed.
• Grzechotka leży pod łóżkiem. = The rattle is lying under the bed.

Also, łóżko is a neuter noun, and in the singular its accusative looks the same as its nominative, so the form stays łóżko.

A often means and, but with a slight sense of contrast, switch of topic, or meanwhile.

So here:

• Smoczek leży obok poduszki, a grzechotka spadła pod łóżko.

sounds like:

• The pacifier is next to the pillow, and meanwhile / while / whereas the rattle fell under the bed.

If you used i, it would sound more like a plain addition of two facts:

• Smoczek leży obok poduszki i grzechotka spadła pod łóżko.

That is possible, but a is more natural when you are talking about one object and then shifting to another.

Because grzechotka is a feminine noun, and in the past tense Polish verbs agree with the subject in gender and number.

So:

• masculine: spadł
• feminine: spadła
• neuter: spadło

Examples:

• Pilot spadł. = The remote fell.
• Grzechotka spadła. = The rattle fell.
• Jabłko spadło. = The apple fell.

Because the sentence describes two different kinds of situations:

• leży = a current state or position
• spadła = a completed event in the past

So the pacifier’s location is being described as it is now:

• Smoczek leży... = The pacifier is lying...

But the rattle’s movement happened earlier:

• grzechotka spadła... = the rattle fell...

This is very natural in Polish, just as it is in English.

It is from the perfective verb spaść.

That matters because perfective verbs usually describe a completed whole event.
So spadła means fell as a finished action.

Compare:

• spaść → spadła = to fall, to have fallen completely
• spadać → spadała = to be falling / to fall repeatedly / to keep falling

In this sentence, the completed event is the important idea, so spadła is the right choice.

Because Polish has no articles.

So:

• smoczek can mean a pacifier or the pacifier
• poduszki can mean a pillow's or the pillow's depending on context
• łóżko can mean a bed or the bed

Polish relies on context instead of separate words like a/an/the.

Not completely. Polish word order is fairly flexible, and changing it often changes emphasis rather than basic meaning.

The given order is neutral and natural:

• Smoczek leży obok poduszki, a grzechotka spadła pod łóżko.

But you could also say:

• Obok poduszki leży smoczek, a pod łóżko spadła grzechotka.

That version puts more emphasis on the locations.

So yes, the original order is natural, but it is not the only possible order.

Because in Polish, a comma is normally used before conjunctions like a, ale, and often before other clause-linking words as well.

Here you have two clauses:

• Smoczek leży obok poduszki
• a grzechotka spadła pod łóżko

So the comma is standard Polish punctuation.

Yes, but it would mean something different.

• pod łóżko = to under the bed, with movement
• pod łóżkiem = under the bed, location only

So:

• Grzechotka spadła pod łóżko. = The rattle fell under the bed.
• Grzechotka jest pod łóżkiem. = The rattle is under the bed.
• Grzechotka leży pod łóżkiem. = The rattle is lying under the bed.

This movement-vs-location distinction is one of the most important things to notice in the sentence.