Η πάνα ήταν στην τσάντα, αλλά η πιπίλα έμεινε πάνω στο τραπέζι.

These are contractions:

• στην = σε + την
• στο = σε + το

This is extremely common in Modern Greek. The preposition σε can mean in, at, to, on, depending on context, and when it is followed by the definite article, it usually contracts like this.

They are in the accusative, because Greek uses the accusative after σε and after expressions like πάνω σε.

So here you have:

• στην τσάντα = σε + την τσάντα
• πάνω στο τραπέζι = πάνω σε + το τραπέζι

For these particular nouns, the noun form looks the same as the nominative, so the article is what makes the case easiest to see.

Because the two location expressions are slightly different:

• στην τσάντα means in the bag
• πάνω στο τραπέζι means on top of the table

Greek often uses σε for general location, but for surfaces, πάνω σε / πάνω στο makes the idea of on top of clearer. So the sentence is contrasting inside the bag with on the table.

The two verbs do different jobs:

• ήταν is the imperfect of είμαι and describes a state: the diaper was in the bag.
• έμεινε is the aorist of μένω and gives the idea of a result: the pacifier stayed / remained / was left on the table.

So the first clause simply gives location, while the second suggests that the pacifier ended up remaining there.

Because Greek nouns have grammatical gender. Both πάνα and πιπίλα are feminine nouns, so they take the nominative singular article η.

This has nothing to do with biological sex. In Greek, objects can be masculine, feminine, or neuter just because that is how the noun is classified. For example, τραπέζι is neuter, so it takes το.

Yes. Greek word order is more flexible than English because articles and endings carry a lot of grammatical information. The given order is neutral and natural, but you could move parts around for emphasis, for example:

Στην τσάντα ήταν η πάνα, αλλά η πιπίλα έμεινε πάνω στο τραπέζι.

That version emphasizes the location in the bag more strongly.

A rough guide:

• πάνα = PA-na
• ήταν = EE-tan
• τσάντα = TSAN-da
• πιπίλα = pee-PEE-la
• έμεινε = E-mi-ne
• τραπέζι = tra-PE-zi

A couple of useful notes:

• τσ sounds like the ts in cats, but at the start of a word.
• The accent mark shows which syllable is stressed.