Learn Hardcore Indonesian: Saya mengatur jadwal ulang tahun keluarga agar tidak bentrok. - I organize the family birthday schedule so it doesn’t clash.

No. Here ulang is part of ulang tahun (birthday). Mengatur ulang means to rearrange. If you meant that, you’d say: Saya mengatur ulang jadwal.

What nuance does mengatur have compared with menjadwalkan or menyusun?

mengatur jadwal: to arrange/manage a schedule (coordinate, adjust times).
menyusun jadwal: to draw up/compile a schedule from scratch.
menjadwalkan [something]: to put something on the schedule (schedule an event). All are fine; choose based on nuance.

Is jadwal ulang tahun keluarga natural, or should I say anggota keluarga?

It’s understandable but a bit ambiguous (a family itself doesn’t have a birthday). Clearer options:

jadwal ulang tahun anggota keluarga (birthdays of family members)
Add possession if relevant: jadwal ulang tahun keluarga kami (our family’s birthdays)

How do I say “our family” vs “my family” here?

keluarga saya = my family
keluarga kami = our family (excluding the listener)
keluarga kita = our family (including the listener)

What does agar do, and can I use supaya or biar?

agar means “so that/in order that.” Alternatives:

supaya: neutral, very common
biar: informal Meaning is the same; register differs.

Why is it tidak bentrok, not jangan bentrok?

tidak negates a verb/adjective/state. jangan is used only for prohibitions/commands. You’re stating a goal (not clashing), not telling someone “don’t clash.”

Do I need to add dengan after bentrok?

Not required. tidak bentrok stands on its own. If you want to specify, use bentrok dengan:

agar tidak bentrok dengan acara lain (so it doesn’t clash with another event)

There’s no subject after agar. Is that okay?

Yes. Indonesian often drops understood subjects. In agar tidak bentrok, the implied subject is the schedule. You can make it explicit: agar jadwalnya tidak bentrok.

How do I show past, ongoing, or future time?

Add particles/adverbs:

Past: Saya sudah mengatur …
Ongoing: Saya sedang/lagi mengatur …
Future: Saya akan mengatur …

What’s a natural casual version? A more formal one?

Casual: Gue ngatur jadwal ultah keluarga biar nggak bentrok.
Formal: Saya menyusun jadwal ulang tahun keluarga agar tidak terjadi benturan jadwal. (also acceptable: agar tidak tumpang tindih)

Can I move parts around?

Yes:

Agar tidak bentrok, saya mengatur jadwal ulang tahun keluarga.
Jadwal ulang tahun keluarga saya atur agar tidak bentrok. (object-fronting for emphasis)

How would a passive version look?

Jadwal ulang tahun keluarga diatur agar tidak bentrok (oleh saya). The oleh-phrase is optional and usually omitted.

Any common shorthand for ulang tahun?

ultah: casual
HUT (Hari Ulang Tahun): formal/official, often for institutions
hari jadi, milad: alternatives in certain regions/contexts

Are there good synonyms for bentrok in this context?

bentrok: most common for schedule clashes
bertabrakan: also used figuratively, a bit stronger/literally “collide”
tumpang tindih: overlap (not necessarily a direct clash), often formal Common collocations: jadwal bentrok, kelas bentrok, agenda bentrok.

Should there be yang between jadwal and ulang tahun?

No. Ulang tahun here is a noun modifier of jadwal. yang introduces a relative clause, not used for simple noun–noun modification.