1. After completing it, please return the form to us. -- Here, on one hand, the phrase 'completing it' seems to be a noun phrase and the 'after' a preposition; on the other hand, the whole "After completing it" seems to be an adverbial phrase. Help. What exactly is it?
A subordinating conjunction introducing an adverbial phrase of time in its use or meaning.
2. She is leaving home after tomorrow. -- Here, 'after' seems to be acting as a preposition with 'tomorrow' acting as a noun.
Yes, it acts as a preposition here. It’s however a subordinating conjunction in "She is leaving home after she finishes high school."
3. Completing it takes a lot of time. -- Here, the phrase 'Completing it' seems to be a noun phrase.
The word ‘completing’ is formed from the verb ‘complete’ by adding -ing after dropping the ‘e’. It functions as a subject of the verb ‘takes’ and it takes an object of its own ‘it’. It does the work of a noun and is therefore a verbal noun or a gerund. ‘Completing it’ is the noun phrase.