Har du lagt verktygen i hallen, eller ligger de kvar i köket?

Because this is a yes/no question. In Swedish, yes/no questions usually put the finite verb first:

• Du har lagt verktygen i hallen. = You have put the tools in the hallway.
• Har du lagt verktygen i hallen? = Have you put the tools in the hallway?

So Swedish often works like English here: Have you... ?

Har lagt is the present perfect.

It is formed with:

• har = the present tense of ha (to have)
• lagt = the supine form of lägga (to put / lay)

So:

• lägga = infinitive
• lägger = present
• lade = past
• lagt = supine, used with har

In this sentence, har lagt focuses on a completed action that is relevant now: Have you put the tools in the hallway...?

Because lagt is the correct supine form of lägga. It is an irregular verb, so you just have to learn the main forms:

• lägga
• lägger
• lade
• lagt

A useful contrast is:

• Har du lagt nycklarna där? = Have you put the keys there?
• Nycklarna ligger där. = The keys are lying there / are there.

They are related, but they describe different things:

• lägga = to put / place something somewhere
  
  • focuses on the action
• ligga = to lie / be lying
  
  • focuses on the resulting position or state

So the sentence contrasts:

• Har du lagt verktygen i hallen...?
  = Did you put them there?
• ...eller ligger de kvar i köket?
  = Or are they still in the kitchen?

This action-vs-state contrast is very common in Swedish.

Kvar means remaining, still there, or left behind.

So ligger de kvar i köket means something like:

• are they still in the kitchen?
• are they still left in the kitchen?

It adds the idea that the tools have not been moved.

You will see kvar in many similar expressions:

• Jag är kvar hemma. = I’m still at home.
• Finns det kaffe kvar? = Is there any coffee left?
• De ligger kvar på bordet. = They’re still lying on the table.

Because de is the subject of ligger.

In ...eller ligger de kvar i köket?, the tools are the ones doing the lying / being located, so de is the subject.

A quick comparison:

• De ligger i köket. = They are in the kitchen.
  (de = subject)
• Jag ser dem i köket. = I see them in the kitchen.
  (dem = object)

In everyday spoken Swedish, both are often pronounced dom, but in writing the distinction is usually kept.

Verktygen is the definite plural form of verktyg (tool).

The noun verktyg is a neuter noun, and its plural pattern is a little unusual because the indefinite singular and plural look the same:

• ett verktyg = a tool
• verktyg = tools
• verktyget = the tool
• verktygen = the tools

So verktygen means the tools.

Swedish often uses the definite form for rooms and locations when talking about a specific, familiar place, especially in a home or other known setting.

So:

• i hallen = in the hallway
• i köket = in the kitchen

This is very natural in Swedish when both speaker and listener know which hallway or kitchen is meant.

English sometimes does the same, but Swedish does it even more regularly in these situations.

Because i means in and is the normal choice for being inside a room or enclosed space.

So:

• i hallen = in the hallway
• i köket = in the kitchen

Swedish often uses:

• i for in
• på for on / at in some other location expressions

For rooms like these, i is the natural choice.

Not always. In Swedish, hallen usually means the entry hall, entrance area, or hallway of a home or building.

So in many cases, hall in Swedish is closer to:

• hallway
• entryway
• entrance hall

rather than every possible meaning of English hall.

Because the second part is describing a current state, not a completed action.

Compare the two halves:

• Har du lagt verktygen i hallen
  = Have you put the tools in the hallway?
  → asks about an action

• eller ligger de kvar i köket?
  = or are they still in the kitchen?
  → asks about their present location/state

So Swedish switches naturally from the present perfect to the present tense.

After eller (or), Swedish starts another clause. Since this second clause is also a question, it keeps the question pattern with the finite verb first:

• ligger = verb
• de = subject
• kvar = adverb
• i köket = place expression

So the order is:

ligger + de + kvar + i köket

That is normal Swedish yes/no question word order.

You could translate it that way word-for-word, but it would sound unnatural in English.

A more natural English translation is:

• Have you put the tools in the hallway, or are they still in the kitchen?

This is a good example of how Swedish and English may use different everyday verbs even when the grammar is closely related.