Po przeprowadzce chcę kupić nowy stół i lampę do salonu.

Because po in this meaning (after) takes the locative case in Polish.

• basic form: przeprowadzka = a move / moving house
• locative singular: przeprowadzce

So:

• po przeprowadzce = after the move / after moving

This is a very common pattern:

• po pracy = after work
• po kolacji = after dinner
• po remoncie = after the renovation

Przeprowadzka means a move, especially moving to a new home/place.

In this sentence, po przeprowadzce means:

• after moving
• after the move
• once I’ve moved

It comes from the verb przeprowadzić się = to move house / relocate.

So the noun and verb are related:

• przeprowadzić się = to move
• przeprowadzka = a move

This is a very normal structure in Polish.

• chcę = I want
• kupić = to buy

Together, chcę kupić means I want to buy.

Polish often uses:

• a conjugated verb first: chcę = I want
• then an infinitive: kupić = to buy

Other examples:

• chcę iść = I want to go
• muszę pracować = I have to work
• lubię czytać = I like to read

Because kupić is the perfective form, and here it fits the idea of buying something successfully / as a completed action.

• kupić = to buy, to purchase (completed result)
• kupować = to be buying / to buy habitually or repeatedly

In this sentence, the speaker means a one-time completed action:

• I want to buy a new table and lamp

So kupić is the natural choice.

Compare:

• Chcę kupić stół. = I want to buy a table.
• Lubię kupować meble. = I like buying furniture.

Because both nouns are objects of kupić, so they are in the accusative case. But different noun types look different in the accusative.

1. stół

This is a masculine inanimate noun.

For many masculine inanimate nouns, the accusative singular looks the same as the nominative singular.

• nominative: stół
• accusative: stół

So after kupić, it stays stół.

2. lampa

This is a feminine noun.

Feminine nouns ending in -a usually change to -ę in the accusative singular.

• nominative: lampa
• accusative: lampę

So:

• kupić stół
• kupić lampę

Because here stół is the direct object of kupić, so it is in the accusative, not the genitive.

For a masculine inanimate noun like stół, the accusative singular looks like the nominative:

• nominative: nowy stół
• accusative: nowy stół

So nowy matches stół in case, gender, and number.

You would see nowego stołu in the genitive, for example:

• Nie ma nowego stołu. = There is no new table.
• Nogi nowego stołu = the legs of the new table

You absolutely can say nowy stół i nową lampę, and that may even be clearer if you want new to apply to both items explicitly.

In the original sentence:

• nowy stół i lampę

the adjective nowy can naturally be understood as applying to both nouns:
a new table and lamp

But Polish, like English, sometimes mentions the adjective only once when it is shared.

Still, if you want to avoid ambiguity, nowy stół i nową lampę is very natural.

Notice the forms:

• nowy stół (masculine accusative = nominative form)
• nową lampę (feminine accusative)

Here it means for the living room.

Literally, do often means to or into, but with furniture and household items, do + genitive is commonly used to mean that something is intended for a room or place.

So:

• stół i lampę do salonu = a table and lamp for the living room

Other examples:

• szafa do sypialni = a wardrobe for the bedroom
• dywan do pokoju = a rug for the room
• krzesła do kuchni = chairs for the kitchen

It does not mean that the speaker wants to physically carry them into the living room right now.

Because do requires the genitive case.

• basic form: salon = living room / lounge
• genitive singular: salonu

So:

• do salonu = for/to the living room

This is a very common preposition-case combination:

• do domu = to home / to the house
• do sklepu = to the shop
• do pokoju = to the room

Yes. In Polish, salon very commonly means living room.

Depending on context, it can also mean salon in the sense of a business establishment, like:

• salon fryzjerski = hair salon
• salon samochodowy = car showroom / car dealership

But in a home-related sentence like this one, salon clearly means living room.

Yes, the word order is natural. The sentence flows well as:

• Po przeprowadzce chcę kupić nowy stół i lampę do salonu.

Polish word order is more flexible than English, because cases show grammatical roles. So other orders are also possible, for example:

• Chcę po przeprowadzce kupić nowy stół i lampę do salonu.
• Nowy stół i lampę do salonu chcę kupić po przeprowadzce.

But the original version is probably the most neutral and natural in everyday speech.

Starting with Po przeprowadzce is especially natural because it sets the time frame first: After the move...

A very natural version would be:

• Po przeprowadzce chcę kupić nowy stół i nową lampę do salonu.

The only important change is the adjective before lampę:

• nową lampę = new lamp (feminine accusative)

So the forms are:

• nowy stół
• nową lampę

This makes it completely clear that both the table and the lamp are new.