Det er den stol, jeg satte mig på, fordi den anden var ustabil.

Because det er is used to identify something specific: it is / that is.

• Det er den stol ... = the speaker is pointing out which chair it was.
• Der er ... means there is / there are and is used to introduce existence, not identification.

So here det er is the natural choice.

In Danish, when a noun is made specific by a following relative clause, it is very common to use den/det/de + noun:

• den stol, jeg satte mig på
• literally: the chair that I sat down on

So den stol here is standard Danish. The den is not necessarily a strong that; in this structure it mainly marks a specific, identifiable chair.

A useful comparison:

• stolen = the chair by itself
• den stol, jeg satte mig på = the chair that I sat down on

Because stol is a common gender noun in Danish.

Its basic form is:

• en stol = a chair

Common gender nouns take den in the definite/article-type forms:

• den stol
• den anden

If it were a neuter noun, you would use det instead.

Because Danish often leaves out the relative pronoun when it is not the subject of the relative clause.

So both of these are possible:

• Det er den stol, jeg satte mig på ...
• Det er den stol, som jeg satte mig på ...

The version without som is very normal and natural.

A useful rule of thumb:

• if the relative word is the subject, Danish usually keeps som or der
• if it is the object or the object of a preposition, it is often omitted

Here the missing relative word would be the thing connected to på, so omission is fine.

This is called preposition stranding, and Danish does it a lot, just like English.

Compare:

• English: the chair I sat on
• Danish: den stol, jeg satte mig på

So the sentence structure is very natural in Danish. The hidden relative element belongs with på, even though på is left at the end.

Because at sætte sig and at sidde mean different things:

• at sætte sig = to sit down / to seat oneself
  
  • change of position
• at sidde = to be sitting
  
  • state or position

So:

• jeg satte mig på stolen = I sat down on the chair
• jeg sad på stolen = I was sitting on the chair

In your sentence, the speaker is referring to the act of choosing that chair and sitting down on it, so satte mig fits well.

Because mig is the object form, and sætte sig is a reflexive verb.

• jeg = subject form = I
• mig = object form = me

So:

• jeg satte mig = literally I sat myself down

That is how Danish expresses to sit down with sætte sig.

Because fordi introduces a subordinate clause.

In standard Danish, subordinate clauses use normal subject-before-verb order:

• fordi den anden var ustabil

This is different from the Danish V2 pattern you see in main clauses.

So after fordi, learners should normally use:

• subject + verb

not a main-clause-style inversion.

You may hear other patterns in casual spoken Danish, but fordi den anden var ustabil is the standard form to learn.

It means the other one, and here it stands for the other chair.

The noun stol is left out because it is already understood from the context.

So:

• den anden = the other one
• more fully: den anden stol = the other chair

This is very common in Danish.

Because it refers to a singular common gender noun: stol.

Forms of anden change with gender and number:

• anden = common gender singular
• andet = neuter singular
• andre = plural

So here:

• en stol
• therefore den anden

If the hidden noun were neuter, you would get det andet.

Because the adjective is describing something understood as a common gender singular noun: stol.

Predicate adjectives in Danish agree like this:

• common gender singular: ustabil
• neuter singular: ustabilt
• plural: ustabile

So here:

• den anden = the other chair
• stol is common gender
• therefore var ustabil

Example comparisons:

• den anden stol var ustabil
• det andet bord var ustabilt
• de andre stole var ustabile

Because the sentence is doing two things at once:

• Det er den stol ... = identifying the chair now
• jeg satte mig på / den anden var ustabil = describing what happened in the past

This is completely normal in Danish, just as in English:

• This is the chair I sat on because the other one was unstable.

So the tense mix is not a problem; it reflects two different time viewpoints.