Certainly an article of some sort is required here, and I personally prefer a to the. But you can easily avoid the problem:
An indicator of a strong character is the ability to compliment the achievements of others.
I think either a or the is required in the context. If you do not use an article , the sentence becomes ungrammatical
An Indicator of a strong character is a / the person' s ability to compliment the achievements of others.
I thiink it can be simplified as:
An indicator of a strong character is a/ the person's ability to ...
If React is the name of the repository, the article may be omitted, though I would probably include it.
If we're talking about a repository that relates somehow to React (e.g. containing it), the article is needed. Consider:
"How do I get to Zabranchburg?” “You can catch the Zabranchburg bus over there.”
You were taught wrong. (Or at least you were taught a rule that works much of the time, but doesn't always).
You use the definite article when you are referring to one specific item. Obviously "the moon" is a good example, but in your case there is (presumably) only one advert with the shirtless guy. Therefore you use the definite article. It doesn't matter ...