Learn Icelandic: Hver er þessi gaur?

When saying this is a [noun], you usually use the neuter þetta er regardless of the noun's gender. However, in this sentence, this is directly attached to the masculine noun gaur (this dude), so it must agree with its gender. That is why it becomes the masculine þessi.

Is gaur considered slang, and when is it appropriate to use?

Gaur is an informal word meaning dude, guy, or bloke. It is extremely common in everyday spoken Icelandic, but you should avoid using it in formal writing or very polite professional contexts. If you need a more neutral word, you would just use maður (man).

Could I ask hvaða gaur er þetta? instead?

You could, but the meaning is a bit different. Hver means who and asks for the person's identity. Hvaða means which or what. Asking hvaða gaur er þetta? translates to which dude is this? and is often used to clarify which person in a group you are talking about.

Does the word hver change depending on the gender of the person I am asking about?