Jeg lægger tråden i skuffen, så jeg ikke mister den.

Because lægge means to lay / to put something somewhere, while ligge means to lie / to be lying somewhere.

• Jeg lægger tråden i skuffen = I put the thread in the drawer.
• Tråden ligger i skuffen = The thread is lying in the drawer.

A useful rule:

• lægge = action you do to an object
• ligge = position/state

This is similar to English lay vs lie, though Danish uses them more regularly.

In Danish, the definite article is usually attached to the end of the noun.

So:

• en tråd = a thread
• tråden = the thread

This is very common in Danish:

• en skuffe = a drawer
• skuffen = the drawer

So tråden does not mean a different word from tråd; it is just the definite form.

i means in, so i skuffen means in the drawer.

That is the natural preposition because the thread is being placed inside the drawer, not on top of it.

Compare:

• i skuffen = in the drawer
• på skuffen = on the drawer

So Jeg lægger tråden i skuffen literally means I put the thread in the drawer.

Here så means so or so that.

It introduces the reason or purpose behind the action:

• Jeg lægger tråden i skuffen = I put the thread in the drawer
• så jeg ikke mister den = so that I do not lose it

In this sentence, så connects the main action with its purpose/result.

Because after så introducing a subordinate clause, Danish normally uses subordinate-clause word order.

That means the order is typically:

subject + sentence adverb + verb + object

So:

• jeg = subject
• ikke = sentence adverb / negation
• mister = verb
• den = object

Hence:

så jeg ikke mister den

This is one of the most important Danish word-order patterns to learn:

• in main clauses, ikke often comes after the finite verb
• in subordinate clauses, ikke usually comes before the finite verb

Because så jeg ikke mister den is a full subordinate clause, and it needs its own subject.

So even though the subject is the same person, Danish still says:

• Jeg lægger tråden i skuffen, så jeg ikke mister den.

Just as in English, you repeat I:

• I put the thread in the drawer so I don’t lose it.

You cannot leave out jeg here.

Because tråd is a common gender noun in Danish:

• en tråd
• therefore the pronoun is den

In Danish:

• nouns with en usually take den
• nouns with et usually take det

So:

• en tråd → den
• et brev → det

In this sentence, den refers back to tråden.

Yes, mister is the present tense of miste.

• at miste = to lose
• mister = lose / am losing / will lose, depending on context

Danish often uses the present tense where English might also use present tense:

• så jeg ikke mister den = so I don’t lose it

This is completely normal. Danish does not need a special future form here.

Sometimes, but the structure changes.

With the same subject, Danish often prefers an infinitive construction:

• Jeg lægger tråden i skuffen for ikke at miste den.

That also means I put the thread in the drawer so as not to lose it.

Your original sentence with så jeg ikke mister den is also correct and natural. The difference is mainly structure:

• så + clause: så jeg ikke mister den
• for at + infinitive: for ikke at miste den

Both are useful to know.

Yes, both can relate to losing, but they are not always interchangeable.

• miste often means lose possession of, be deprived of, or misplace
• tabe often means drop or lose in the sense of losing a game, competition, or control

In this sentence, miste is natural because the meaning is so I don’t lose/misplace it.

Examples:

• Jeg mistede mine nøgler. = I lost my keys.
• Jeg tabte bolden. = I dropped the ball.
• Vi tabte kampen. = We lost the match.

Because Danish often uses commas to separate the main clause from a following subordinate clause.

Here:

• main clause: Jeg lægger tråden i skuffen
• subordinate clause: så jeg ikke mister den

So the comma helps show the structure of the sentence.

Comma rules in Danish can be a bit different from English, but in a sentence like this, the comma is very normal.

Then you would use ligger, not lægger:

• Tråden ligger i skuffen. = The thread is in the drawer / The thread is lying in the drawer.

So the difference is:

• Jeg lægger tråden i skuffen. = I put the thread in the drawer.
  → action
• Tråden ligger i skuffen. = The thread is in the drawer.
  → state/location

This contrast is very important in Danish.