Learn Hardcore German: Ich trage die Kette heute im Büro. - I am wearing the necklace at the office today.

You would only see der Kette if Kette were in the dative or genitive singular (e.g. mit der Kette, wegen der Kette).
Das Kette is simply wrong, because Kette is not neuter.

Which case is die Kette in here, and how can I tell?

In Ich trage die Kette heute im Büro, die Kette is in the accusative case.

Reason: die Kette is the direct object – it is the thing being worn/carried.

Ich = subject (nominative)
trage = verb
die Kette = direct object (accusative)
heute im Büro = adverbials (time and place)

For feminine nouns, the nominative singular and accusative singular both use die, so you have to rely on the sentence role (subject vs. object) to know the case.

Why is the verb form trage and not trag or trägt?

The verb is tragen (to wear / to carry), and it’s irregular in the stem vowel for some persons. Present tense conjugation:

ich trage (I wear / I am wearing)
du trägst (you wear – singular, informal)
er / sie / es trägt (he / she / it wears)
wir tragen (we wear)
ihr tragt (you wear – plural informal)
sie / Sie tragen (they wear / you wear – formal)

Because the subject is ich, you must use trage.
Trag by itself is just the stem, not a correct finite form here.

Does tragen mean “to wear” or “to carry”? How is it normally used?

Tragen can mean both, depending on context:

to wear (clothes, jewelry, accessories)
- Ich trage die Kette. – I am wearing the necklace.
- Er trägt einen Anzug. – He is wearing a suit.
to carry (objects, people)
- Ich trage die Tasche. – I am carrying the bag.
- Sie trägt das Kind. – She is carrying the child.

In your sentence with die Kette, the natural meaning is to wear, because necklaces are normally worn, not carried.

You can also use anhaben for clothes/jewelry in spoken German:

Ich habe die Kette an. – I have the necklace on / I’m wearing the necklace.

Both tragen and anhaben are common; tragen sounds a bit more neutral/formal, anhaben is more colloquial.

German has no “-ing” form like English. Does Ich trage die Kette mean “I wear the necklace” or “I am wearing the necklace”?

It can mean both. German does not have a separate progressive tense like English.

Ich trage die Kette heute im Büro.
= I am wearing the necklace at the office today.
or, in the right context,
= I wear the necklace at the office today (e.g. as part of a schedule).

Usually, context makes it clear that this is about what is happening today, so English will typically translate it as a progressive: I am wearing…

Why do we say im Büro and not in dem Büro or in das Büro?

Im is a standard contraction in German:

im = in dem

So im Büro literally is in dem Büro (“in the office”), just in shortened, more natural form.

About in das Büro:

in dem Büro / im Büro → location (where something happens) → dative case
- Ich arbeite im Büro. – I work in the office.
- Ich trage die Kette heute im Büro. – I am wearing the necklace in the office today.
in das Büro → movement into the office → accusative case
- Ich gehe in das Büro. / Ich gehe ins Büro. – I am going into the office.

So in your sentence we are talking about where you are wearing the necklace (location), not about moving somewhere, so im Büro (dative) is correct.

Which case does Büro take after im, and what is the gender of Büro?

Im = in dem, and dem is the dative singular form of the neuter article das.

So:

Gender: das Büro (neuter noun)
Dative singular: dem Büro
Contracted: im Büro = in dem Büro

In this sentence, Büro is therefore neuter dative singular.

Could I also say Ich trage heute die Kette im Büro or Heute trage ich die Kette im Büro? Are those correct?

Yes, both are correct. German word order is flexible, especially for adverbials (time, place, etc.), and you can move elements for emphasis.

Some natural variants:

Ich trage die Kette heute im Büro.
– neutral; slight focus on die Kette as the object.
Ich trage heute die Kette im Büro.
– time (heute) comes directly after the verb; fairly neutral and very common.
Heute trage ich die Kette im Büro.
– puts strong emphasis on Heute (“Today [and not on other days] I’m wearing the necklace in the office”).

All three are grammatical. Choice depends on what you want to emphasize:

Emphasize time: start with Heute.
Emphasize contrast of object: stress die Kette in speech.

Is there any rule for the order of heute and im Büro? Why not Ich trage die Kette im Büro heute?

There is a common guideline for the order of adverbials in German: Te-Ka-Mo-Lo (Temporal, Kausal, Modal, Lokal = Time, Reason, Manner, Place).

So the typical order is:

Time (heute)
Manner / other adverbs
Place (im Büro)

Applied to your sentence, the most neutral order of adverbs is:

… heute im Büro.

Ich trage die Kette im Büro heute is not wrong, but it sounds slightly less natural and may sound like you are emphasizing heute at the end for contrast or clarification. In normal, neutral speech, Germans would usually say … heute im Büro.

Why is the article die used with Kette, but no article is used with Büro after im?

Büro does have an article; it’s just hidden inside the contraction im.

Basic forms:
- die Kette – the necklace
- das Büro – the office

In the sentence:

die Kette – article is visible (accusative feminine)
im Büro – article is inside im = in dem, so dem is the article (dative neuter)

So both nouns actually have definite articles here:

die Kette
in dem Büro → im Büro

Would it be more natural to say Ich werde die Kette heute im Büro tragen if I am talking about the future?

You can say Ich werde die Kette heute im Büro tragen, and that is grammatically correct future tense (Futur I).

However, in everyday German, the present tense is usually used for near future plans when the time is clear from context:

Ich trage die Kette heute im Büro.
– I’m going to wear the necklace in the office today.

Use werde + Infinitiv mostly when:

you want to emphasize the future aspect strongly, or
the time is not clear from context, or
you are making a prediction.

For a simple plan “today at the office”, the present tense is perfectly normal and probably more common.

Can Kette mean only “necklace”, or also “chain”?

Kette has both meanings; context decides:

Necklace (jewelry)
- Sie trägt eine goldene Kette. – She is wearing a gold necklace.
Chain (metal links, bike chain, etc.)
- Die Kette des Fahrrads ist kaputt. – The bike chain is broken.
- Die Hunde waren an Ketten. – The dogs were on chains.

In Ich trage die Kette heute im Büro, the natural interpretation is necklace, because you normally don’t wear a metal chain in an office unless it is jewelry. Context will make this clear to the listener.