Learn Hardcore French: Selon la caissière, ce sac coûte moins cher ailleurs. - According to the cashier, this bag costs less elsewhere.

They’re interchangeable in this sentence: D’après la caissière, ce sac coûte moins cher ailleurs.
Nuance: selon is neutral/formal. d’après can also mean “based on” (evidence) or “after/in the style of” (art), but with a person as a source it’s fine.

Why is it la caissière? What’s the masculine form, and is gendering job titles normal?

Masculine: le caissier; feminine: la caissière (note the accent: è).
Many French job titles have masculine/feminine forms (e.g., vendeur/vendeuse, serveur/serveuse).
If you don’t want to specify gender: in speech you might say la personne à la caisse; in writing you often see forms like le/la caissier(-ère).

Why the definite article la (and not une) before caissière?

la caissière refers to a specific, identifiable cashier (e.g., the one at this store right now).
une caissière would be “a cashier” in general, not necessarily the one involved in the situation. It’s possible, but it shifts the meaning.

Why is it ce sac and not cet sac or cette sac?

ce = masculine singular before a consonant: ce sac.
cet = masculine singular before a vowel or mute h: cet homme, cet hôtel.
cette = feminine singular: cette robe.
ces = plural for both genders: ces sacs, ces robes.

Does coûte need the circumflex? Is coute also correct?

Since the 1990 spelling reforms, the circumflex on û is optional in many words: coûte and coute are both accepted.
In practice, coûte/coûter/coût are still commonly written with the circumflex in standard writing.
Pronunciation doesn’t change: coûte is [kut].

Why is it moins cher and not moins chère/moins chers?

With coûter, cher functions adverbially (“expensively”), so it’s invariable:

Ça coûte (très) cher / plus cher / moins cher. If you use être, cher is an adjective and agrees:
Ce sac est cher. Ces robes sont chères.

Could I say est moins cher instead of coûte moins cher?

Yes. Both are idiomatic:

Ce sac est moins cher ailleurs.
Ce sac coûte moins cher ailleurs. Using coûter focuses more explicitly on the price, but both are common.

What’s the difference between moins cher ailleurs and moins cher qu’ailleurs?

They say opposite things:

moins cher ailleurs = it’s cheaper somewhere else (implies here is more expensive).
moins cher qu’ailleurs = here it’s cheaper than elsewhere (often used with ici: Ici, ce sac coûte moins cher qu’ailleurs.)

What exactly does ailleurs mean here, and how is it different from d’ailleurs or autre part?

ailleurs = “elsewhere / somewhere else.” No preposition needed: coûte moins cher ailleurs.
d’ailleurs usually means “besides / moreover / incidentally” in discourse. Don’t use it for the price comparison here.
autre part also means “elsewhere,” but ailleurs is more idiomatic in this context.

Where can I place selon la caissière in the sentence, and do I need the comma?

Placement is flexible; use commas to set it off:

Any pronunciation tips for this sentence?

Selon [səlɔ̃]; la [la]; caissière [kɛsjɛʁ]; ce [sə]; sac [sak]; coûte [kut]; moins [mwɛ̃]; cher [ʃɛʁ]; ailleurs [ajœʁ].
The final -s in moins is silent. The -t in coûte is pronounced.
A natural liaison is often made between cher and ailleurs: [ʃɛʁ‿ajœʁ].
Whole sentence: [səlɔ̃ la kɛsjɛʁ, sə sak kut mwɛ̃ ʃɛʁ‿ajœʁ].

Can I say moins de cher or plus bon marché?

No: with adjectives, use moins/plus + adjective without de: moins cher, plus cher.
bon marché means “inexpensive/cheap” and is invariable: Ce sac est bon marché.
Its comparative is traditionally meilleur marché (not plus bon marché), but in everyday speech moins cher/plus cher is far more common.

How do I compare prices with a specific place, like “cheaper than at Carrefour” or “cheaper here”?

“cheaper than at Carrefour”: moins cher qu’à Carrefour.
“cheaper here”: moins cher ici or Ici, c’est moins cher.
Other examples: moins cher en ligne / sur Internet, moins cher à Paris, moins cher que chez X.

Is there a difference between coûter and valoir in this context? What about revenir (à)?

coûter = what you have to pay: Ce sac coûte 50 €.
valoir = to be worth (value): Ce sac vaut 50 €. Often interchangeable with coûter for prices, but coûter is the go-to in pricing contexts.
revenir à = to come to/amount to (for the buyer), often after discounts: Avec la réduction, ça revient à 45 €.

Could I replace caissière with another word like vendeuse?

caissière = cashier (works at the till).
vendeuse = sales assistant/shop clerk (on the sales floor). If the person is specifically at the register, caissière is the precise term; in small shops roles can overlap, but choose according to the job.