Learn Hardcore German: Neben dem Fenster steht ein bequemer Sessel, und dahinter brennt eine Kerze. - Next to the window there is a comfortable armchair, and behind it a candle is burning.

dem Fenster is in the dative case. The preposition neben can take either dative or accusative, depending on whether it expresses location (dative) or direction/movement (accusative). Here we’re talking about the location “next to the window,” so neben governs the dative.

Why does the sentence start with Neben dem Fenster, and how does that affect word order?

German is a V2 (verb-second) language. When you front a prepositional phrase (Neben dem Fenster) for emphasis or style, the finite verb (steht) must occupy the second position. The subject (ein bequemer Sessel) follows the verb. In English we’d say “Next to the window a comfortable armchair stands,” but we’d normally invert that.

Why is there a comma before und in … Sessel, und dahinter …? I thought commas before “and” in German were optional.

Correct, commas before coordinating conjunctions like und are generally optional in German. However, a comma is often used to clarify when each clause has its own subject and verb. Here you have two independent clauses:
1) … steht ein Sessel
2) … brennt eine Kerze
Putting a comma before und makes that division clearer, though you could omit it in everyday writing.

Why does bequemer end in -er? I expected something like bequeme or bequemes.

bequemer is an adjective modifying Sessel, which is masculine nominative singular. The indefinite article ein only shows that it’s masculine but doesn’t fully mark case. Therefore the adjective takes the strong ending -er to signal masculine nominative. So you get ein bequemer Sessel.

What kind of word is dahinter, and why not just say hinter dem Sessel?

dahinter is a locative adverb (a pronominal adverb) combining da- + hinter (“there-behind”). It replaces the prepositional phrase hinter dem Sessel and makes the sentence more concise. Both are correct:
• und dahinter brennt eine Kerze.
• und hinter dem Sessel brennt eine Kerze.

Why is there an indefinite article eine before Kerze, but none before Sessel at the very start?

Actually, there is also an indefinite article before Sessel: ein. Both nouns use an indefinite article because we’re introducing them for the first time and neither is uniquely identified. We have:
• ein bequemer Sessel
• eine Kerze

What exactly does brennt mean here? Does it mean “burns” as in “on fire”?

In this context brennen means “to be lit” or “to burn (with flame),” i.e. the candle is giving off light. It doesn’t mean “to burn up” or “to be destroyed by fire,” but simply that the wick is alight.

Could I say Neben dem Fenster liegt ein bequemer Sessel instead of steht? How would the meaning change?

Yes, you could, but it would imply a different posture of the armchair. stehen is used for upright objects standing on their feet/legs. liegen would suggest something is lying down. Since chairs “stand,” steht is the natural verb. If you used liegt, it would sound odd—as if the chair were on its side or very low.