Grammar of ‘More Carefully Than I Do’ in Comparative Sentences

Does the author clarify further in the passage that 'complement' can be part of a clause?

To take your last, main question first, No, Prof. Eastwood does not make this clear in the snippet you provide. However, it may have been made clear earlier, or it may be made clear later, after Prof. Eastwood has laid what he considers an adequate foundation.

What is meant by 'other phrases, too' here? Does it mean other sentence elements, like complement? Or does it mean the types of phrases other than 'noun phrase' and 'verb phrase'.

It appears that Prof. Eastwood is building his introduction to syntax on the phrase, so I imagine that he means phrases of all types, and that he uses the term phrase in a fairly narrow sense. His approach is by no means universal in grammatical discourse; verb phrase, for instance, in many grammars denotes not just the few words will be but "the entire string of words governed (or headed) by a verb". Your quotation from Wikipedia employs 'verb phrase' in the latter sense. Traditional grammar, on the other hand, calls what Prof. Eastwood identifies as a 'verb phrase' simply the verb.

So as you can see, the matter is complicated by critical terminological differences; and we haven't yet even started to address what a clause is!

I am not acquainted with the Oxford Guide to English Grammar, so I cannot be sure; but it seems to me that Prof. Eastwood is trying to build your house from the ground up, brick by brick, rather than exhibiting its overall design first, and only then showing you where the bricks fit. I therefore suggest that you would do better to take his study in larger chunks. Read through an entire chapter to get a sense of his argument; then go back and reread, and perhaps everything will be clearer. If anything still baffles you you will at least be able to bring us specific, narrow questions we can handle within the 200-800 words of an ordinary answer.