Die Brötchen backe ich kurz im Ofen, damit sie warm sind.

I briefly bake the bread rolls in the oven so that they are warm.

Breakdown of Die Brötchen backe ich kurz im Ofen, damit sie warm sind.

• sein: to be
• in: in
• dem: the; (masculine or neuter, dative)
• ich: I
• warm: warm
• sie: they
• damit: so that
• backen: to bake
• das Brötchen: the bread roll
• kurz: briefly
• der Ofen: the oven

Questions & Answers about Die Brötchen backe ich kurz im Ofen, damit sie warm sind.

Why is the finite verb backe in second position and the subject ich after it, even though ich is the doer of the action?

In German main clauses the finite verb must occupy the V2 (second) position. Here Die Brötchen (the direct object) are fronted for emphasis or topic, so backe stays in slot two and the subject ich follows. If you began with the subject instead, you’d say Ich backe die Brötchen… but fronting the object shifts ich behind the verb.

What part of speech is kurz and what does it modify? Why is it placed before im Ofen?

kurz is an adverb of time/duration (“for a short time”). It modifies the action backe … im Ofen, telling you how long you bake the rolls. Adverbs in German typically come after the verb (and subject) but before other complements such as prepositional phrases, hence backe ich kurz im Ofen.

Why do we say im Ofen instead of in den Ofen?

in is a two-way preposition. When indicating location (answering “wo?”), German uses the dative case: in dem Ofen, contracted to im Ofen. in den Ofen (accusative) would express motion into the oven, but here you’re simply describing where the baking happens.

Why is damit used here, and how is it different from um … zu?

damit is a subordinating conjunction meaning “so that,” introducing a clause with its own finite verb (sie sind). You typically use damit when the subject of the purpose clause differs from the main clause subject (here sie, “they,” versus ich).
um … zu also expresses purpose but uses an infinitive phrase and requires the subject to be the same. Example:
  Ich backe die Brötchen kurz im Ofen, um sie warm zu machen.

Why do we use warm sind instead of warm sein or warm werden in the subordinate clause?

warm sind is the finite present tense of sein for sie (“they are warm”), describing the desired resulting state.
• warm sein would be an infinitive (“to be warm”) and needs zu if used in a purpose phrase.
• warm werden (“to become warm”) emphasizes the process of heating.
Since the focus here is on the result—so that the rolls end up warm—you say damit sie warm sind.

How can I tell that die Brötchen is a direct object and not the subject, since die looks the same in nominative and accusative plural?

Plural die serves for both nominative and accusative, so you rely on syntax and verb argument structure. ich is the clear subject (I bake), and backen takes a direct object (what is baked). Therefore die Brötchen functions as the accusative object. If die Brötchen were the subject, the verb would be backen (they bake), not backe.

Is it possible to change the word order and still keep the same meaning?

Yes. You can say:
  Ich backe die Brötchen kurz im Ofen, damit sie warm sind.
Or even start with the purpose clause:
  Damit sie warm sind, backe ich die Brötchen kurz im Ofen.
The core meaning stays the same; you only shift emphasis or topic by rearranging clauses.