Learn Icelandic: Hvers konar skrautgripur er þetta?

It is technically two words because hvers is the genitive (possessive) form of hver (what/which), and konar comes from an old noun meaning "sort" or "kind". While similar phrases are sometimes combined into a single word (like allskonar, meaning "all kinds"), hvers konar is generally written as two separate words.

Does "hvers konar" need to change to match the masculine gender of "skrautgripur"?

No, it doesn't. Genitive type descriptors like -konar act as frozen, indeclinable adjectives. Even though skrautgripur (ornament) is masculine, hvers konar will stay exactly the same regardless of the noun's gender, case, or whether it is singular or plural.

The lesson also mentions "-s lags". Could I use that instead in this sentence?

Yes! You could absolutely say Hvers lags skrautgripur er þetta? The phrases hvers konar and hvers lags mean exactly the same thing ("what kind of") and are completely interchangeable in everyday speech.

Since "skrautgripur" is masculine, why does the sentence end with the neuter pronoun "þetta" instead of the masculine "þessi"?