Learn Dutch: Onze nieuwe machine is heel efficiënt. - Our new machine is very efficient.

The word machine is a de word. For de words, we always use the possessive onze (not ons, which is for het words). Because it is a de word, the adjective describing it also gets an extra -e at the end, making it nieuwe.

Why does efficiënt have two dots over the second 'e'?

Those two dots are called a trema. In Dutch, the letters i and e together normally make an 'ee' sound (like in niet). The dots on the ë tell you to break the vowel combination and pronounce the e as a new, separate syllable.

How would the word order change if I started this sentence with Bovendien (Moreover)?

Because bovendien is an adverb, putting it at the start of the sentence triggers verb-second inversion. The verb is must stay in the second position, so the subject onze nieuwe machine moves right after it. You would say: Bovendien is onze nieuwe machine heel efficiënt.

Could I use erg instead of heel in this sentence?