Der Plan ist geheim.

Because Plan is a masculine noun in German. German nouns have grammatical gender, and Plan happens to be masculine, so in the nominative singular it takes der.

• der Plan = masculine
• die would be for a feminine noun
• das would be for a neuter noun

This is something you usually have to learn together with the noun: not just Plan, but der Plan.

It is nominative case.

In Der Plan ist geheim, der Plan is the subject of the sentence, and the subject is normally in the nominative.

So:

• der Plan = nominative subject
• ist = verb
• geheim = predicate adjective describing the subject

Because all nouns are capitalized in German.

So:

• der Plan
• das Haus
• die Idee

This is a standard spelling rule in German, not something special about this sentence.

Because German main clauses usually follow the V2 rule: the finite verb comes in the second position.

In this sentence:

• Der Plan = first position
• ist = second position
• geheim = rest of the clause

So the structure is very normal German word order: Subject + verb + complement

Because geheim is being used after the verb ist, not directly before a noun.

In German, adjectives usually take endings only when they come before a noun:

• der geheime Plan = the secret plan

But after sein and similar verbs, the adjective is in its basic form:

• Der Plan ist geheim.

So:

• geheime = before a noun
• geheim = after ist

It is an adjective.

More specifically, it is a predicate adjective because it comes after the verb sein and describes the subject.

Compare:

• der geheime Plan → adjective before a noun
• Der Plan ist geheim. → predicate adjective after sein

It is not an adverb in this sentence.

Yes, but that is not a full sentence by itself in the same way.

• Der Plan ist geheim. = a complete sentence
• der geheime Plan = a noun phrase, meaning the secret plan

So they are related, but they do different jobs:

• one is a full statement
• the other is just a phrase

Normally, no. In standard German, singular countable nouns usually need a determiner such as an article.

So:

• Der Plan ist geheim. = normal
• Ein Plan ist geheim. = also possible, depending on meaning
• Plan ist geheim. = generally not standard in this context

German usually requires something like der, ein, dieser, etc. with a singular noun like Plan.

The difference is the article:

• Der Plan = the plan, a specific plan already known in the context
• Ein Plan = a plan, one plan among others, not specifically identified yet

So the grammar is the same, but the meaning changes depending on whether the noun is definite or indefinite.

Yes. It is a completely natural and standard sentence.

The pattern is very common:

• Das ist gut.
• Der Film ist interessant.
• Das Haus ist groß.
• Der Plan ist geheim.

It is a basic German sentence structure: subject + sein + adjective.