Learn Italian: La stima è controllata dal supervisore, altrimenti facciamo uno sbaglio. - The estimate is checked by the supervisor, otherwise we make a mistake.

No, it doesn't. Altrimenti (otherwise) introduces a factual consequence of what happens if the first action isn't taken, so it almost always takes the indicative mood. That's why we use the regular present tense facciamo here.

Can I put altrimenti at the very beginning of the sentence instead?

Not in this context. Because altrimenti acts as a connector of alternative, it needs a premise to react to. It must bridge the initial condition (La stima è controllata...) with its consequence (...facciamo uno sbaglio).

Why does controllata end in -a here?

Because the first clause uses the passive voice: "The estimate is checked". In passive constructions formed with the verb essere, the past participle acts like an adjective and must agree in gender and number with the subject. Since la stima is feminine singular, controllato becomes controllata.

I thought stima meant "respect" or "esteem". Does it also mean "estimate"?

Yes, it frequently carries both meanings! In interpersonal contexts, it means "esteem" (as in avere molta stima di qualcuno). However, in business, project management, or finance, una stima is the standard word for "an estimate" or "an appraisal".