Learn Hardcore Dutch: Na het eten vullen we de thermos opnieuw met heet water. - After eating we refill the thermos with hot water.

Na het eten is a time adverbial meaning “after eating” or “after the meal.” Here eten is a noun derived from the verb eten (“to eat”), referring specifically to the act or occasion of eating. In Dutch, this noun is neuter, so it takes the definite article het.

Why does the verb vullen appear before the subject we in this sentence?

This is due to inversion. When you start a Dutch main clause with any element other than the subject (for example, a time phrase like Na het eten), the verb immediately follows that fronted element, then comes the subject. Structure:

Fronted element (Na het eten)
Verb (vullen)
Subject (we)

What does opnieuw mean, and how does it compare to weer?

Both opnieuw and weer translate as “again.”

Opnieuw emphasizes repeating the entire action from the start (“fill it again from empty”).
Weer is more colloquial and general.
In this sentence you could also say:
“We vullen de thermos weer met heet water,”
but opnieuw often sounds a bit more formal or precise.

Why is it de thermos and not het thermos?

Many Dutch nouns borrowed from other languages (especially those ending in -us) are assigned the common gender, so they use de. Thermos in Dutch is treated as a common-gender noun: de thermos.

Why is there no article before heet water?

Water is an uncountable noun when you speak about it in general. In such cases Dutch typically omits the article:

Correct: vullen we de thermos met heet water
With an article (to specify): vullen we de thermos met het hete water uit de flessenwarmer

Why is heet spelled with a double e, and why doesn’t it get an -e ending here?

The double e indicates a long vowel sound, pronounced /e:/.
Adjectives in Dutch take an extra -e only when they precede a definite or plural noun (de + noun, het with non–neuter, or any plural). Here water is uncountable and used without an article, so the adjective remains in its base form: heet water, not hete water.

Why is vullen used without any preposition? What about opvullen?

Vullen is a transitive verb that takes a direct object (“fill something”). You say vullen de thermos met water without adding a preposition to the verb itself.
Opvullen is a separable verb meaning “to fill up (completely).” You could say:

“Na het eten vullen we de thermos weer op.”
but then op moves to the end: vullen … op.

Can I change the word order to put the subject first, for example We vullen na het eten de thermos opnieuw met heet water?

Yes. If you keep the subject we in first position, you do not invert. The verb stays in second position:
“We vullen na het eten de thermos opnieuw met heet water.”