Learn Dutch: De man geeft zuchtend het antwoord. - The man gives the answer, sighing.

zuchtend is the present participle. You make it by taking the full infinitive verb zuchten (to sigh) and adding -d. Here, it acts as an adverb describing how the man gives the answer (he does it sighing or with a sigh).

Why is there no -e at the end of zuchtend?

Because it is being used as an adverb describing the verb geeft, not an adjective. If you were using it as an adjective right in front of a noun, like the sighing man, you would add an -e and say de zuchtende man. But as an adverb, it stays just zuchtend.

Why does zuchtend come before het antwoord instead of at the end?

In Dutch, adverbs describing how something happens (adverbs of manner) generally come after the main verb but before the direct object. So the Dutch word order places zuchtend right after geeft, unlike English which often puts the -ing word at the very end of the sentence.

Could I translate the start of this sentence as "The man is giving..."?

Yes, Dutch uses the simple present tense geeft for both gives and is giving. Using a simple verb like geeft combined with the present participle zuchtend is a very common and elegant way in Dutch to show two things happening at once.