Tom snijdt een wortel en een champignon op de snijplank.

Because the subject is Tom, which is third-person singular (he). The infinitive is snijden (to cut), but in the present tense Dutch changes the verb:

• ik snijd = I cut / I am cutting
• jij snijdt = you cut
• hij snijdt = he cuts
• Tom snijdt = Tom cuts

So snijdt is the correct present-tense form with Tom.

This is a very common Dutch spelling point.

The stem of snijden is snijd. For third-person singular present tense, Dutch usually adds -t to the stem:

• stem: snijd
• • t
• result: snijdt

Even though it is written -dt, it is normally pronounced just like t at the end.

Een is the indefinite article, meaning a or an.

So:

• een wortel = a carrot
• een champignon = a mushroom

In Dutch, een does not change for gender, so both de-words and het-words use een in the singular when they are indefinite.

Because the sentence is talking about one carrot and one mushroom.

Dutch often repeats the article when two singular countable nouns are listed:

• een wortel en een champignon

This is very natural and clear. If you leave out the second een, it may sound less precise or less natural in many beginner contexts.

Because snijplank is a de-word. In Dutch, singular nouns usually take either de or het, and this has to be learned with the noun.

So:

• de snijplank = the cutting board

There is no simple rule that lets you predict every noun correctly, so it is best to learn nouns together with their article.

That is normal Dutch word order. In a simple main clause, Dutch often goes:

subject + finite verb + object + place/time/etc.

So here:

• Tom = subject
• snijdt = finite verb
• een wortel en een champignon = object
• op de snijplank = place

Putting the location at the end is very natural:

• Tom snijdt een wortel en een champignon op de snijplank.

No. Here op is just a preposition meaning on.

So:

• op de snijplank = on the cutting board

It is not a separable verb such as opsnijden. The main verb here is simply snijden.

It is a compound noun:

• snijden = to cut
• plank = board

So snijplank literally means something like cutting board.

Dutch uses compound nouns very often, and they are usually written as one word.

Yes. Champignon is a normal Dutch word, borrowed from French. In everyday Dutch it usually means the common edible mushroom, especially the button mushroom.

Its plural is:

• champignons

So:

• een champignon = a mushroom
• twee champignons = two mushrooms

A learner-friendly approximation is something like snite.

A few useful points:

• the sn- at the beginning is pronounced
• ij is a Dutch vowel sound, often roughly compared to ei in some accents of English, though it is not exactly the same
• the final dt is not pronounced as two separate sounds; it sounds like a final t

So although it is spelled snijdt, the ending sounds simple.

You can use this pattern:

Subject + verb + object + place

For example:

• Tom snijdt een appel op de snijplank.
• Sara snijdt brood in de keuken.

So the sentence is a very useful model for making similar Dutch sentences.

Yes. That is another common way to express an ongoing action.

Compare:

• Tom snijdt een wortel en een champignon op de snijplank.
  = simple present; very normal in Dutch

• Tom is een wortel en een champignon aan het snijden op de snijplank.
  = emphasizes that he is in the middle of cutting them

Dutch uses the simple present more often than English does, so the original sentence is completely natural.

Not always. Wortel can also mean root in some contexts.

But in an everyday food sentence like this one, een wortel is understood as a carrot. Context usually makes the meaning clear.