Learn Hardcore Indonesian: Rencana tim ditulis singkat agar semua orang paham. - The team plan is written briefly so that everyone understands.

Yes. ditulis is the passive form of menulis (“to write”). The prefix di- marks passive voice with the patient as subject. The agent (the writer) is omitted, which is common in Indonesian.

You can add the agent with oleh: Rencana tim ditulis singkat oleh manajer. (The team plan was written briefly by the manager.)
Or use the “short passive” (passive type 2): Rencana tim saya tulis singkat. (I wrote the team plan briefly.)

Can I rephrase it in active voice?

Yes:

Tim menulis rencana itu secara singkat agar semua orang paham. (The team wrote the plan briefly so everyone understands.)
With a specific agent: Manajer menulis rencana tim secara singkat agar semua orang paham.

Is "singkat" an adjective or an adverb here? Do I need "secara singkat"?

Here singkat functions adverbially (“briefly”), modifying ditulis. Indonesian often uses adjectives as adverbs of manner.

ditulis singkat and ditulis secara singkat both work; the latter is a bit more formal.
Position: put it after the verb for manner. Rencana tim singkat would mean “a brief team plan” (adjective modifying the noun), not “written briefly.”