Gramatical terminology varies from country to country and I have never heard some of the terms in your list. However, as I see it, most - if not all - of your answers are probably correct. I have never heard of "regular adjectives" but
sunny certainly is an adjective. "Pronominal adjective" beats me; there's no such thing in the terminology I am familiar with. Perhaps they want
their, which, as far as I understand, is called an adjective in some countries. We always call it a possessive pronoun, which isn't a very good name either because it doesn't replace a noun.
For "preposition - group of words" - I would say in front of, but it isn't in the text. And even then front is a noun, and therefore the term seems unpalatable to me. So is a co-ordinating conjunction in the text and when is a subordinating conjunction.
Beckoning is a present participle in the text, but taking, the first word, is a gerund. Your infinitives are right. Wish (May not wish) is an example of an infinitive without to, called either a plain infinitive or a bare infinitive owing to the absence of the particle to.
Markers is an appositive. The expletives are correct. The term predicate is used completely differently in Scandinavia from the Anglo-Saxon world, and my idea of what it is is the same as yours but I don't think your teacher wants that if he/she wants the English version. I won't venture a guess at all. The same goes for "complete subject", which baffles me.
I hope my comments help you. Perhaps a native speaker who knows the terms used in the Anglo-Saxon world can provide more information and better answers.
CB