Learn Hardcore Indonesian: Kabel pengisi daya dia pendek, jadi kami pindah ke stopkontak dekat meja. - His/Her charger cable is short, so we move to the power outlet near the table.

It’s grammatical and understandable.
Many speakers prefer kabel pengisi dayanya (his/her charging cable) because -nya feels smoother.
Very common colloquial options: kabel charger dia or kabel chargernya.
Formal/neutral but less common: kabel pengisi daya miliknya.
Spelling tip: write chargernya (no hyphen), and pengisi dayanya (suffix attached to the last word of the phrase).

Why is there no adalah before pendek?

With adjective predicates, Indonesian normally omits adalah: Kabel ... pendek (not adalah pendek).
Use adalah mainly before nouns/pronouns: Dia adalah guru.

Should it be kami or kita here?

kami = we (excluding the listener).
kita = we (including the listener).
Pick kami if the person you’re talking to didn’t join the move; use kita if they did.

How is jadi working here? Do I need the comma? Any alternatives?

jadi = so/therefore, marking a result or conclusion.
Comma is optional: ..., jadi ... or Jadi, ... at the start for emphasis.
Alternatives: sehingga, karena itu (therefore), and colloquial makanya.

Is pindah the right verb, or should it be berpindah or memindahkan?

Why ke stopkontak and not di stopkontak?

Is stopkontak the usual word? What about colokan or soket?

Should it be dekat meja, di dekat meja, or yang dekat meja?

All are acceptable; choose by clarity and style:
- dekat meja = near the table (concise, common in speech).
- di dekat meja = near the table (clear and standard).
- yang dekat meja = the one that is near the table (specifies which outlet).

Is pengisi daya really used, or do people just say charger?

pengisi daya is the official/neutral term.
In daily conversation, charger is extremely common: kabel charger, chargernya.
Verbs: standard mengisi daya, colloquial ngecas (to charge).

Does dia mark gender? How do ia, beliau, and -nya compare?

dia is gender-neutral (he/she).
ia is a more formal subject pronoun in writing; uncommon after nouns or prepositions.
beliau is respectful/polite for elders or superiors.
-nya can mean his/her/its, and can also mark definiteness: kabelnya = the cable / his/her cable (context decides).

Is Kabel pengisi dayanya pendek better style than Kabel pengisi daya dia pendek?

Many speakers find the -nya version smoother and slightly more neutral: Kabel pengisi dayanya pendek.
The dia version is fine and common in casual speech, especially with charger: Kabel charger dia pendek.

If I want to say we plugged it in rather than we moved, how can I say that?

Colloquial: Kami colok di/ke stopkontak dekat meja.
More precise: Kami mencolokkan kabel ke stopkontak dekat meja or Kami menancapkan steker ke stopkontak.

How do I show past time? There’s no tense ending like -ed in Indonesian, right?

Correct—no tense inflection. Use time words:
- tadi kami pindah ... (earlier)
- barusan kami pindah ... (just now)
- kemarin kami pindah ... (yesterday)

Is pendek the right adjective here? When would I use singkat or tidak cukup panjang?

Is the spelling stopkontak one word?

Yes. Standard spelling is stopkontak (one word). You will see stop kontak informally, but stopkontak is the dictionary form.