Jeg skærer mere ost, mens min veninde finder flere ingredienser frem.

In modern Danish present tense, most verbs have a single present form (often ending in -r) that’s used with all subjects:

• jeg skærer, du skærer, han skærer, vi skærer, de skærer
  The infinitive is at skære. There’s no separate -s / -es like English cuts / cut.

Jeg skærer is present tense and can cover both English I cut and I am cutting, depending on context. Danish doesn’t have a dedicated continuous form like English; the simple present often does that job.

Ost is often treated like an uncountable “mass” noun (like cheese in English), so you can say mere ost without an article.
If you mean a specific unit (a cheese, a cheese wheel, a piece), then you might use an article or another noun:

• en ost = a (whole) cheese
• et stykke ost = a piece of cheese

• mere = more (amount, uncountable or “mass”) → mere ost
• flere = more (number, countable plural) → flere ingredienser
  So the sentence nicely contrasts “more cheese (amount)” with “more ingredients (additional items).”

Mens commonly means while (simultaneous actions), which fits here. It can also mean whereas (contrast) in other contexts. Usually you can tell from meaning and context; here it’s clearly about timing.

Danish typically uses a comma to separate clauses. Here you have:

• Main clause: Jeg skærer mere ost,
• Subordinate clause introduced by mens: mens min veninde finder flere ingredienser frem.
  So the comma marks the clause boundary.

After mens, you’re in a subordinate clause, and Danish subordinate clauses usually have subject + verb order:

• mens min veninde finder ...
  In main clauses, Danish often uses V2 word order (verb in second position), but the subordinate clause pattern is different.

• en ven = a (male) friend / sometimes gender-neutral in casual use
• en veninde = a female friend
  So min veninde specifies that the friend is female.

Yes, Danish possessives agree with the grammatical gender/number of the noun:

• min (common gender singular): min veninde, min ven
• mit (neuter singular): mit hus
• mine (plural): mine venner, mine ingredienser

finde frem is a particle verb meaning something like to get out / to bring out / to produce (from storage) depending on context.
Danish often splits particle verbs:

• min veninde finder flere ingredienser frem
  The main verb finder comes earlier, and the particle frem tends to appear later (often near the end of the clause).

You can, but it changes the meaning:

• finder flere ingredienser = finds (discovers/locates) more ingredients
• finder flere ingredienser frem = gets (takes) more ingredients out (e.g., from a cupboard/fridge)
  With cooking context, frem is very natural.