Learn Hardcore German: Die Lehrerin meint, das Treffen sei beispielsweise am Freitag möglich. - The teacher says the meeting is possible on Friday, for example.

Because this is reported speech (indirekte Rede). German marks reported statements with the subjunctive I (Konjunktiv I). The Konjunktiv I of sein is sei. It shows you are relaying the teacher’s statement rather than asserting it as your own fact.

Direct speech: Die Lehrerin sagt: Das Treffen ist am Freitag möglich.
Reported speech: Die Lehrerin meint, das Treffen sei am Freitag möglich.

Can I just use ist here? What changes?

You’ll hear it, but it changes the nuance:

Die Lehrerin meint, das Treffen ist am Freitag möglich. This sounds like the writer/speaker is endorsing the statement as fact (indicative).
Die Lehrerin meint, das Treffen sei am Freitag möglich. This clearly marks it as the teacher’s claim (reported speech), leaving the reporter neutral. In careful or formal writing, prefer sei. In casual speech, the indicative after reporting verbs is common but less precise.

Why isn’t the verb at the end of the clause like in other subordinate clauses?

This is an “unintroduced” content clause of reported speech (uneingeleiteter Nebensatz). Such clauses often keep main-clause word order (verb in second position, V2), especially with Konjunktiv I:

V2 in reported speech: …, das Treffen sei … If you introduce the clause with dass, then the finite verb moves to the end:
…, dass das Treffen … möglich sei.

Can I add dass? If so, what happens to the verb form?

Yes:

Die Lehrerin meint, dass das Treffen beispielsweise am Freitag möglich sei. Using dass is very common. In formal writing, many still keep Konjunktiv I (sei) after dass to mark reported speech. Using ist after dass implies the writer treats it as fact:
Die Lehrerin meint, dass das Treffen … möglich ist. (reporter aligns with the content)

Is the das in das Treffen the same as dass?

No. das here is the neuter definite article for the noun Treffen. dass (with double s) is the conjunction “that.” In this sentence the conjunction dass is omitted, but the article das is required:

Correct: …, (dass) das Treffen …
Wrong: …, dass Treffen … (missing article)

What exactly does beispielsweise mean, and how does it compare to zum Beispiel or etwa?

beispielsweise = “for example,” a bit more formal/neutral in written German.
zum Beispiel (often abbreviated z. B.) = the most common equivalent; slightly more colloquial than beispielsweise, but fine in most contexts.
etwa can mean “for example” but also “approximately,” so it’s potentially ambiguous. In your sentence, etwa would be understood as “for example.”

Where else could I put beispielsweise?

Acceptable placements (subtle shifts in emphasis):

Neutral: …, das Treffen sei beispielsweise am Freitag möglich.
Slightly emphasizing Friday: …, das Treffen sei am Freitag beispielsweise möglich.
Fronted focus on the “for example”: …, beispielsweise sei das Treffen am Freitag möglich. Avoid moving it so it modifies the reporting verb, unless that’s your intent:
Die Lehrerin meint beispielsweise, … = “The teacher, for example, thinks …” (scope changes)

Is am Freitag the only option? What about Freitag or freitags?

am Freitag = on Friday (a specific Friday) — neutral and very common.
Bare Freitag (no preposition) can work adverbially in many contexts: …, das Treffen sei Freitag möglich. This sounds a bit more colloquial.
freitags = on Fridays (habitually, every Friday): …, das Treffen sei freitags möglich. That changes the meaning to a general habit. Don’t say am Freitags — that’s incorrect.

Why is there a comma after meint even though there’s no dass?

German requires a comma before a subordinate content clause, even if it’s unintroduced (no dass). The comma signals the clause boundary:

Die Lehrerin meint, das Treffen sei …

Does meinen mean “to say” or “to think” here? Could I use sagen, glauben, finden, or denken instead?

meinen here is “to be of the opinion/claim,” a bit softer than sagen but commonly used with reported speech.
Alternatives:
- sagen (to say) — neutral report: Die Lehrerin sagt, …
- glauben (to believe) — belief rather than assertion: …, sie glaube, …
- finden (to think/feel, informal): …, sie findet, …
- denken (to think) — less common to introduce content clauses in this exact pattern, but possible: …, sie denkt, … All can take reported speech; with careful writing, keep Konjunktiv I in the content clause.

Why not wäre instead of sei?

wäre is Konjunktiv II. In reported speech, German prefers Konjunktiv I (sei). Use Konjunktiv II only when the Konjunktiv I form is identical to the indicative and could cause confusion (not the case with sein, because sei clearly differs from ist). Using wäre here can sound either colloquial or like you’re adding tentativeness that isn’t in the original.

Can I use the würde-construction (e.g., würde … möglich sein)?

Yes, in everyday speech/writing you’ll see it as a workaround for the subjunctive:

Die Lehrerin meint, das Treffen würde am Freitag möglich sein. It’s acceptable informally. In formal writing, prefer Konjunktiv I (sei).

Is sei … möglich the best way to express possibility? Could I say könne … stattfinden?

Both are fine; they differ slightly in focus:

…, das Treffen sei am Freitag möglich. Stresses possibility in general.
…, das Treffen könne am Freitag stattfinden. Uses the verb stattfinden (to take place) and Konjunktiv I of können (könne), highlighting the event’s ability to occur. Both are idiomatic in reported speech.

Why are Lehrerin, Treffen, and Freitag capitalized?

German capitalizes all nouns, including abstract/event nouns like Treffen and weekdays like Freitag. Lehrerin is a noun (female teacher), so it’s capitalized. The adverb beispielsweise is lowercase.

What happens if I change meint to the past (meinte)?

You keep Konjunktiv I for simultaneity in reported speech:

Die Lehrerin meinte, das Treffen sei beispielsweise am Freitag möglich. If you need past-before-past (anteriority), use the perfect in Konjunktiv I:
…, das Treffen sei am Freitag möglich gewesen. (had been possible)