Learn German: Welchen Zweck hat dieser auffällige Gürtel? - What purpose does this flashy belt have?

Because the strong masculine nominative marker is already on the demonstrative pronoun dieser. When the preceding word already clearly shows the gender and case, the following adjective takes the weak ending, which is just -e for the nominative case.

The current lesson is about the suffixes -haft and -mäßig. Why is this adjective formed with -ig (auffällig) instead?

The suffixes -haft and -mäßig are almost always attached to nouns (like Held becoming heldenhaft). However, auffällig is derived from the verb auffallen (to stand out). When creating adjectives from verbs, German typically uses suffixes like -ig, -lich, or -bar, which often trigger an umlaut change on the root vowel (from a to ä).

Could I ask Was für einen Zweck instead of Welchen Zweck in this context?

Yes, both work well, though they have a slight nuance. Welchen Zweck literally asks 'Which purpose?' and sounds slightly more specific, as if there is a particular function the belt is meant to serve. Was für einen Zweck translates closer to 'What kind of purpose?' and sounds slightly more open-ended.