Learn Dutch: Jouw eerste argument is erg overtuigend. - Your first argument is very convincing.

Since this sentence is about evaluating debate points, using the emphatic jouw (your) rather than the unstressed je highlights that it is exactly your argument being praised, possibly in contrast to someone else's.

Since argument is a het word, why does eerste get an -e ending?

Possessive pronouns like jouw make the noun definite, which triggers the -e ending on adjectives for both de and het words. Furthermore, ordinal numbers like eerste (first) almost always take an -e ending when placed before a noun anyway.

Could I use heel instead of erg before overtuigend?

Yes, heel overtuigend means exactly the same thing. While learners often first encounter erg with negative things (like erg koud or erg duur), it is perfectly natural to use it with positive words too, just like 'very' in English.

How is the adjective overtuigend formed?

It is a present participle. By adding a -d to the infinitive of the verb overtuigen (to convince), you create an adjective that means 'convincing'.