Learn Hardcore German: Wir haben denselben Plan, also bleiben wir ruhig.

denselben is the accusative masculine form of derselbe (the very same / the exact same).
Because Plan is the direct object of haben, it must be in the accusative: Wir haben … den Plan → Wir haben denselben Plan.

der gleiche Plan means the same kind of plan / an identical plan, but not necessarily the exact one.

derselbe Plan = the one and only exact plan we both share
der gleiche Plan = a plan of the same type (could be two separate but identical plans)

Also, den gleichen Plan would be accusative too, but with the meaning an identical plan, not the exact same one.

How do I know Plan is masculine, and how does that affect denselben?

der Plan is masculine in German. You can see it in the dictionary entry der Plan.
Masculine nouns take these forms of derselbe:

nominative: derselbe Plan
accusative: denselben Plan
dative: demselben Plan
genitive: desselben Plans

Here you need the accusative, so you get denselben.

Why does derselbe turn into denselben? That seems like a lot of change.

derselbe declines like the definite article der/die/das plus an adjective ending:

article part changes (der → den in masculine accusative)
adjective ending is added: -en in masculine accusative

So: der + selb- + e (base) becomes den + selb- + en → denselben.

What does also mean here—does it mean “also” as in “too”?

No. German also usually means so / therefore / in that case.
In this sentence: Wir haben denselben Plan, also bleiben wir ruhig. = “We have the same plan, so we stay calm.”

If you want “also/too” in German, you’d usually use auch.

Is the comma before also required?

Yes, because also introduces a new main clause here:

Wir haben denselben Plan (main clause 1)
also bleiben wir ruhig (main clause 2)

German typically separates main clauses with a comma, especially when there’s a connector like also, aber, denn, deshalb, trotzdem.

Why is the word order also bleiben wir ruhig and not also wir bleiben ruhig?

In a main clause, the finite verb must be in position 2 (V2 rule).
When also is placed first, it counts as position 1, so the verb comes next:

Also (position 1) bleiben (position 2) wir …

Also wir bleiben ruhig is possible in speech, but it’s more like a discourse filler (“So, we’re staying calm…”) and less like the neat logical connector meaning “therefore.”

What exactly does ruhig bleiben mean? Is it “quiet” or “calm”?

ruhig bleiben usually means to stay calm / keep calm (not necessarily silent).
ruhig can mean both quiet and calm, but with emotions or reactions it’s typically calm.

If you specifically mean “stay quiet (don’t make noise),” you might say leise sein/bleiben or still sein/bleiben depending on context.

Why is it bleiben wir and not wir bleiben?

Because also is in the first position of the clause, the verb must come second. That pushes the subject after the verb:

Also bleiben wir ruhig.

If the subject is first, then you get:

Wir bleiben ruhig.

Both are correct; the difference is what you put in the first position (topic/focus).

Can I replace also with deshalb or darum? Would the sentence change?

Yes, you can, and the grammar stays the same (verb still second):

…, deshalb bleiben wir ruhig.
…, darum bleiben wir ruhig.

Nuance:

also = “so/therefore” (often neutral, common in speech and writing)
deshalb = “for that reason” (a bit more explicit/logical)
darum = “that’s why” (often slightly more conversational)

Is Wir haben denselben Plan something Germans would actually say, or is there a more natural alternative?

It’s correct and natural. Depending on context, Germans might also say:

Wir haben den gleichen Plan. (often used in everyday speech, even if technically “identical” rather than “the exact same”)
Wir sind uns einig. (“We agree.”)
Wir verfolgen denselben Plan. (“We’re pursuing the same plan.”)

If you truly want to emphasize “the exact same one,” denselben is the precise choice.

Could Plan be plural here? What would it look like?

Yes, if you meant “the same plans,” you’d say:

Wir haben dieselben Pläne, also bleiben wir ruhig.

Plural forms of derselbe:

nominative plural: dieselben Pläne
accusative plural: dieselben Pläne (same form as nominative plural)
dative plural: denselben Plänen