Learn Hardcore German: Wir merken den Fehler sofort. - We notice the mistake immediately.

Because den is the accusative singular article for masculine nouns. Fehler is masculine (der Fehler in the nominative), but as the direct object of merken it takes accusative: den Fehler. Masculine: nominative der → accusative den.

Does merken take the accusative or dative?

In the sense of “notice,” merken is a transitive verb that takes a direct object in the accusative: etwas merken (Akk.). The dative shows up only in the separate reflexive construction sich (Dat.) etwas merken (“to memorize/remember something”).

Can I say Wir merken uns den Fehler sofort? How does that change the meaning?

Yes, but it means something different: Wir merken uns den Fehler sofort = “We commit the mistake to memory right away.” Without uns, Wir merken den Fehler sofort means “We notice the mistake right away.”

Where should sofort go? Is Wir merken sofort den Fehler also correct?

Neutral and very common: Wir merken den Fehler sofort.
Also possible (slightly different rhythm/focus): Wir merken sofort den Fehler.
With fronted time for emphasis: Sofort merken wir den Fehler.
In a subordinate clause: …, dass wir den Fehler sofort merken. Pronouns tend to come before short adverbs: Wir merken ihn sofort (not so natural: “Wir merken sofort ihn”).

How do I negate this? Where does nicht go?

“Not immediately”: Wir merken den Fehler nicht sofort.
“Not at all”: Wir merken den Fehler nicht. (You can add emphasis: gar nicht.)

What’s the difference between merken, bemerken, auffallen, erkennen, feststellen, and entdecken?

merken: neutral “notice/perceive”; very common in speech. Takes Akk.: etwas merken or a clause: merken, dass …
bemerken: a bit more formal/explicit “notice.” Very natural with mistakes: einen Fehler bemerken.
auffallen: the thing is the subject; the experiencer is dative: Uns (Dat.) fällt der Fehler (Nom.) auf. Feels like “it strikes us.”
erkennen: “recognize/realize (identify correctly).” E.g., Wir erkennen unseren Fehler; or Wir erkennen, dass …
feststellen: “ascertain/determine (often after checking).” Wir stellen den Fehler fest.
entdecken: “discover (often after searching).” Wir entdecken den Fehler.

Is the past tense Wir haben den Fehler sofort gemerkt correct? What about Wir merkten den Fehler sofort?

Both are correct.

Spoken German prefers the perfect: Wir haben den Fehler sofort gemerkt.
Simple past is fine, more written in many regions: Wir merkten den Fehler sofort.

Can I front the object for emphasis: Den Fehler merken wir sofort?

Yes. German is verb-second, so you can put the object first for focus: Den Fehler merken wir sofort. This highlights the object.

Can I replace den Fehler with a pronoun?

Yes: Wir merken ihn sofort. With pronouns, this order (pronoun before the time adverb) is preferred.

Is the passive Der Fehler wird sofort gemerkt acceptable?

Grammatically yes, but it sounds unusual. Idiomatic choices are:

Der Fehler wird sofort bemerkt.
Der Fehler wird sofort festgestellt. Or with auffallen: Der Fehler fällt sofort auf.

Why not dem Fehler? When would dem appear?

dem Fehler is dative. You’d need a verb or preposition that requires dative. Example with auffallen: Uns (Dat.) fällt der Fehler (Nom.) sofort auf. With merken (notice), the mistake is a direct object, so accusative: den Fehler.

What are the gender and plural of Fehler?

der Fehler (masculine). Plural: die Fehler (no change in the noun). Examples:

Singular: den Fehler merken
Plural: die Fehler merken or in general statements: Fehler merken

When do I use den vs einen here?

den Fehler = a specific, known mistake.
einen Fehler = some (unspecified) mistake. Both are accusative masculine: den (definite), einen (indefinite).

Can I use man for a general statement?

Yes: Man merkt den Fehler sofort. This means “One/people notice the mistake right away.” Verb stays 3rd person singular with man.

Is merken regular? How is it conjugated?

Yes, it’s regular and not separable.

Present: ich merke, du merkst, er/sie/es merkt, wir merken, ihr merkt, sie/Sie merken
Perfect: haben gemerkt
Simple past: merkten
Past participle: gemerkt

Any pitfalls with sofort vs gleich?

sofort is unambiguous “immediately.” gleich can mean “immediately” or “in a moment/soon,” depending on context and region. If you want to be clear, prefer sofort.

Pronunciation tips for the sentence?

Wir [viːɐ̯]
merken [ˈmɛʁkən] (in many accents the final -en is lightly reduced)
den [deːn]
Fehler [ˈfeːlɐ] (the h marks a long e)
sofort [zoˈfɔʁt] (stress on the second syllable)