See this sentence "This is my last night working here"! What grammar structure is it? Reduced Relative Clause?

Question

We often reduce relative clauses when the relative pronoun is the subject of relative clause.

The professor who is teaching mathematics will leave the university. Reduced: The professor teaching mathematics will leave the university. (source)

"who is teaching mathematics" is the relative clause & the relative pronoun "who" is subject of relative clause.

But, I am not sure if the sentence "This is my last night working here" (Jennifer Aniston said in the Friend, at 0:40) can be interpreted as "This is my last night that I work here" & people removed "that I" & changed "work" to "working" or something like that???.

Also, see other example from the dictionary,

I will be busy cleaning the house. (source)

Why do they put a gerund right after an adjective like that?

Note: thoughtco says

The present participle is sometimes used as an adverb to describe the manner in which a verb is performed.

She taught pounding the grammar into their heads!

Angelo works considering all angles.

Notice how the present participle could be preceded with 'by' to give the same meaning:

She taught (by) pounding the grammar into their heads!

Angelo works (by) considering all angles.

Answer 1

There are two elements in the gerund-participial which are understood through context:

(subject) working here (temporal location)

The subject of working is inferred from my.

The temporal location is interpreted through the head noun night, which in this case could not possibly be the subject of working, and so is understood as the temporal location of the situation expressed by working here.

A similar example with a wh-relative clause where an adjunct of temporal location has been relativized would be:

I would like to say a word or two on the subject of this Amendment before we pass from it, as this is the last night when we shall discuss the King's Speech (Mr. William Watson, House of Commons, 1937)

Which could be paraphrased as:

our last night discussing the King's Speech

Though this version would make it seem that the discussion takes up most of the evening, whereas the original simply locates the discussion (however brief) on that evening.

Gerund-participials as post-head modifier in NPs are very similar to relative clauses, but are not analyzed as relative clauses because it is not possible for them to include a relative phrase. (The Cambridge Grammar of the English Language p1265)

*our last night when discussing the kings speech

Other instances requiring a similar interpretation:

When faced with challenges, Donaldson is fond of telling people that on his first day working at the State Department in 1973 as undersecretary for energy policy, war broke out in the Middle East. (Bush picks polar opposite of Pitt to take over at SEC, Greg Farrell, USA Today 2002)

She'd snuck those shoes out of her mother's closet moments after May Jones left for her PTA meeting, and prayed the whole hour and a half while her mother was gone that she would hurry back and be so dead tired from her long day teaching third grade and arguing with the parents of Spruce Street Elementary School that she'd go straight to sleep without making a big fuss over what time Deena turned the lights out, and wouldn't come back in the house looking for those shoes. (Dreamgirls; Millner, Denene; Harper Collins 2006)

The title, " Ms. Bixby's Last Day, " refers to her last day teaching at the school and the inevitability of one's last day living. (Book review: 'Mrs. Bixby's Last Day' a tale of life, loss for classmates, Virginian-Pilot 2019)

It was like watching a parade of past Julias: a kittenish little number she'd worn during her year singing a cappella; a pleated skirt and sweater set that had been her daily uniform as a temp. (Julia and Sunny, Sarah Shun-lien Bynum, Ploughshares 2018)

His years working in France as a cook had left him with culinary skills far greater than one would suspect from the three-table restaurant he ran with his guest room. (Each Generation Offers Its Kindness; Robert M. Press; Christian Science Monitor 1992)

Answer 2

This is my last night working here.

looks like a combination between

This is my last night here AND This is the last night (that) I am working here.

as if my replaced the omitted (that) I am. The sentence could be re-written as (I am overdoing it, for the sake of analysis):

This is the last night you see me working here OR This is the last night of my working here.

Another very common phrase of this kind is one's last time doing something:

To pay for lodging for his horses during the winter, he worked in the Washington woods one summer, his last time doing so.

The conclusion is that people do use these structures, though it is difficult to find grammatical regulations or explanations for them.

Busy + ing is a totally different story. Wikipedia says that in this case the gerund is

Part of adjective phrase

• They are all [busy eating cakes].

Cambridge defines busy and shows how it can be used:

busy adjective (DOING THINGS)
If you are busy, you are working hard, or giving your attention to a particular thing:

• Mum was busy in the kitchen.
• The kids are busy with their homework.
• She's busy writing out the wedding invitations.

So, if you really want to stretch it you could see busy cleaning as busy [with] cleaning (in which case the gerund would be used as a noun).
Others have suggested in or while instead of with:

I will be busy [while] cleaning the house.

Here, busy cleaning the house is seen by certain grammarians as a case of Adjective complementation by an -ing participle clause.

It does seem though that this structure has become much more common especially after the turn of the third millennium which is interesting.

Conclusion: You will find examples with busy doing something in most dictionaries. It is a correct idiomatic phrase which is particular because it has, as you pointed out, an adjective followed by the -ing form of a verb (without a preposition unlike bad at joking or afraid of diving).

Answer 3

This a reduced relative clause for sure, but your interpretation is incorrect, the reduced relative clause for This my last night __ working here is not:

This is my last night that I work here

BUT:

This is my last night (that I am) working here

Why? Because the verb "be" can be omitted in reduced relative clauses, I will cite this rule (look at (d) for a similar example):

2. If the relative clause contains the verb 'be' + any of the following then it can be reduced:

(a) be + an adjective phrase

• The man (who is) interested in your car will telephone later.

• The man interested in your car will telephone later.

Note that other verbs are possible here as well as 'be', such as 'seem', 'look' and 'appear'.

(b) be + a prepositional phrase

• The books (which are) on the table have been read.

• The books on the table have been read.

(c) be + a past participle [a passive form]

• A person (who has been) tricked once is careful the next time.

• A person tricked once is careful the next time.

(d) be + a present participle [a continuous form]

• The family (who are) living in the house are very rich.

• The family living in the house are very rich.

Source: https://learnenglish.britishcouncil.org/grammar/intermediate-to-upper-intermediate/relative-clauses-defining-relative-clauses