Kan du ge mig skruvmejseln från verktygslådan och lägga hammaren på bordet?

Because Swedish forms yes/no questions by putting the finite verb before the subject.

• Du kan = You can as a statement
• Kan du = Can you as a question

In real use, Kan du ...? is often not just about ability. It is also a normal way to make a request, just like in English. So here it means something like a polite Could you ...?

Because they come after the modal verb kan.

After modal verbs such as kan, ska, vill, måste, Swedish normally uses the infinitive without att:

• kan ge
• kan lägga

So:

• Kan du ge ... och lägga ...?

not

• Kan du ger ... och lägger ...?

This works much like English can give and can put, not can gives and can puts.

Because mig is the object form of jag.

• jag = I
• mig = me

In this sentence, the person is receiving something, so Swedish uses the object form:

• ge mig skruvmejseln = give me the screwdriver

Compare:

• Jag ger dig ... = I give you ...
• Du ger mig ... = You give me ...

In Swedish, with a verb like ge, it is very common to put the indirect object pronoun before the direct object noun:

• ge mig skruvmejseln

This is the most natural order here.

Other patterns are possible, but they sound different:

• ge skruvmejseln till mig = also possible, with more emphasis on to me
• ge skruvmejseln mig = generally unnatural in modern everyday Swedish

So for learners, ge mig något is a very useful pattern to remember.

In Swedish, the definite article is usually attached to the end of the noun instead of being a separate word.

So:

• skruvmejsel = screwdriver

• skruvmejseln = the screwdriver

• verktygslåda = toolbox

• verktygslådan = the toolbox

• hammare = hammer

• hammaren = the hammer

• bord = table
• bordet = the table

The endings depend partly on the noun’s gender:

• common gender nouns often take -en or -n
• neuter nouns often take -et or -t

Here, bord is neuter, so it becomes bordet.

Yes. Swedish uses compound nouns very freely, much more than English does.

• verktyg = tool
• låda = box
• verktygslåda = toolbox
• verktygslådan = the toolbox

And:

• skruv = screw
• mejsel = chisel / driver
• skruvmejsel = screwdriver
• skruvmejseln = the screwdriver

A very common Swedish pattern is:

1. make a compound noun
2. then add the definite ending at the end of the whole word

Från means from, and it works well here because the screwdriver is being taken from the toolbox.

• från verktygslådan = from the toolbox

You could also hear ur verktygslådan, which means more literally out of the toolbox. That can sound a bit more specific about something coming from inside it.

So:

• från verktygslådan = natural and common
• ur verktygslådan = also possible, with a stronger sense of out of

Both are understandable, but från is broad and very common in everyday speech.

Because på means on.

• på bordet = on the table

This is the normal preposition when something is placed on top of a surface.

So the second part means that the hammer should end up on the table.

Swedish often uses different verbs for putting things somewhere, depending on position.

• lägga = lay something down, so it lies flat
• ställa = set something so it stands upright
• sätta = put something into a seated/fixed position
• placera = place, more neutral or formal

A hammer placed on a table will normally be lying down, so lägga is the natural choice.

That is why Swedish says:

• lägga hammaren på bordet

rather than using ställa, which would suggest an upright position.

Because both infinitives, ge and lägga, are linked to the same modal verb kan and the same subject du.

So the structure is basically:

• Kan du ge ... och lägga ...?

This means:

• Can you give ... and put ...?

Swedish does not need to repeat du or kan before the second verb. The same subject carries over naturally.

In most everyday contexts, it will be understood as a request, not a real question about whether the person is physically capable.

So it works like English:

• Can you give me the screwdriver ... and put the hammer on the table?

If you wanted to sound a bit softer or more polite, Swedish might also use:

• Skulle du kunna ge mig skruvmejseln från verktygslådan och lägga hammaren på bordet?

But the original sentence is completely normal.