Learn Hardcore Malay: Kami mengelak orang ramai dengan keluar melalui pintu keluar belakang. - We avoid the crowd by exiting through the back exit.

Kami = we (not including you, the listener). That’s what the sentence implies.
Kita = we (including you). Use this if you’re telling the listener they were part of the action. Example: Kami sudah pergi. (We went, but not you.) vs Kita sudah pergi. (We, including you, went.)

What exactly does Orang ramai mean? How is it different from Ramai orang or Orang-orang?

Orang ramai is a fixed phrase meaning “the public” or “the crowd” (a collective).
Ramai orang means “many people” (focus on quantity).
Orang-orang is a marked plural “people,” often stylistic/emphatic; it’s not the usual way to say “many people.” Examples:
Orang ramai diminta bertenang. = The public/crowd is asked to remain calm.
Ramai orang menunggu. = Many people are waiting.

Is Mengelak orang ramai correct, or should it be Mengelakkan orang ramai or Mengelakkan diri daripada orang ramai?

Mengelak + [noun] is fine and common when you “avoid/dodge” something tangible: mengelak orang ramai, mengelak kereta, etc.
Mengelak daripada + [noun/clause] is also very natural, especially in careful/formal Malay: mengelak daripada orang ramai.
Mengelakkan + [noun] is fully transitive and often used with undesirable outcomes: mengelakkan kemalangan, mengelakkan kesesakan. If you say mengelakkan orang ramai it can sound like “to make the crowd avoid (something)” because -kan can be causative. To say you kept yourself away, the idiomatic form is mengelakkan diri daripada orang ramai. Natural options here:
Kami mengelak orang ramai …
Kami mengelak daripada orang ramai …
Kami mengelakkan diri daripada orang ramai … (most explicit)

How does the pattern Dengan + verb work here (dengan keluar)?

Dengan + verb means “by V‑ing / using the method of V‑ing.” So dengan keluar = “by going out.” Other common patterns:

dengan cara + verb phrase: more explicit “by the method of…”
dengan menggunakan + noun: “by using…” Examples:
Dia menjimatkan masa dengan mengambil teksi.
Mereka menyelesaikan masalah itu dengan berbincang.

Should I use Daripada or Dari after Mengelak?

Use daripada with people/abstract sources; use dari with places/time/directions.

Correct: mengelak daripada orang ramai
Also common in speech but less formal: mengelak dari orang ramai Compare:
Keluar dari bilik (from a place)
Belajar daripada guru (from a person)

Keluar melalui vs keluar dari vs ikut: which preposition fits best?

melalui = through/via (route). Keluar melalui pintu … emphasizes the path used.
dari = out of/from (origin). Keluar dari pintu … emphasizes motion out of a doorway.
ikut = via/along/through (colloquial). Keluar ikut pintu … is very natural in speech.
menerusi = through/via (formal alternative to melalui). All are acceptable with slight nuance differences; your original uses melalui, which is good.

What does Pintu keluar belakang mean exactly, and how is it different from Pintu belakang?

Pintu keluar = an exit door (often as signed/labelled).
Pintu belakang = the back door/rear door (not necessarily a designated exit).
Pintu keluar belakang = the rear exit (an exit door located at the back). You can also say pintu keluar di belakang for clarity.

Is repeating Keluar twice (the verb and pintu keluar) awkward?

It’s grammatical and acceptable. If you want to avoid repetition:

Kami mengelak orang ramai dengan menggunakan pintu keluar belakang.
Kami mengelak orang ramai dengan keluar melalui pintu belakang.
Kami mengelak orang ramai dengan keluar ikut pintu belakang.

Why is the order Pintu keluar belakang and not Belakang pintu keluar?

Malay is head-first: noun head followed by descriptors.

pintu (head) + keluar (type: exit) + belakang (location: rear) = “rear exit door.”
belakang pintu keluar would mean “the back part of the exit door,” which is different. You can also say pintu keluar di belakang (“exit door at the back”) if you prefer a prepositional phrase.

Is Keluar transitive? How is it different from Mengeluarkan?

keluar is intransitive (“to go out/exit”): it doesn’t take a direct object. It pairs with a preposition: keluar melalui/dari/ikut [place].
mengeluarkan is transitive (“to take/bring out; to issue/produce”): Dia mengeluarkan telefon, Syarikat itu mengeluarkan produk baharu.
Imperative keluarkan! = “take it out!”

Can I drop the subject pronoun Kami?

In casual speech, yes—if context is clear. You might hear something like Mengelak orang ramai dengan keluar … But in standard writing and to avoid ambiguity (and to signal exclusive we), keep Kami.

Are there good synonyms for Mengelak here?

Yes:

menghindari (more formal/standard transitive “to avoid”): menghindari orang ramai
menjauhkan diri (daripada) (“to keep oneself away from”): menjauhkan diri daripada orang ramai
mengelakkan diri (daripada) (very idiomatic): mengelakkan diri daripada orang ramai

Any differences if I say this in Indonesian?

Indonesian would typically use:

menghindari (avoid) rather than mengelak
kerumunan (crowd) or orang banyak rather than orang ramai
lewat (via) is common in Indonesian but not in Malaysian Malay for this meaning A natural Indonesian version would be along the lines of: Kami menghindari kerumunan dengan keluar lewat pintu belakang.

Does Orang ramai mean “the crowd” or “a crowd”? Malay has no articles—how do I make it specific?

Malay has no the/a. Orang ramai can be specific or generic from context. To emphasize “a crowd,” you can say:

sekumpulan orang / satu kumpulan orang To emphasize definiteness, use context or add a limiter, e.g., orang ramai di lobi (“the crowd in the lobby”).

Any quick pronunciation tips for key words?

mengelak: muh-NGEH-lak (the e is a schwa; ng = [ŋ])
keluar: kuh-LOO-ar (two syllables “ku-luar,” the u is like “oo”)
melalui: muh-lah-LOO-ee
orang ramai: OH-rang rah-MAI (ai as in “eye”) Malay stress is light and usually near the end; keep vowels clear and pronounce every syllable.