Are there any examples of people using verbs such as いただく, 参る, 申す, いたす (and their 尊敬語 counterparts, along with the various other humble/respectful verbs) in those plain forms, rather than conjugated as 丁寧語? I also mean at the end of a sentence, rather than as part of a larger clause that requires dictionary forms.
丁寧語 is about who you are talking to. Its use is to be polite to your listener/reader.
尊敬語/謙譲語 are about who you are talking about. These patterns show respect to the agent. 謙譲語 to lower yourself (or someone on the same side as you vis-a-vis your listener) and 尊敬語 to raise the person you're talking about.
Thus, you can say
when speaking to a friend to simultaneously respect the teacher and communicate casually with a friend.
To kind of complete the pattern,
makes sense to say to a teacher about something that happened with a peer.
I can't imagine when you could say 申す at the end by itself considering for its meaning...
I can think of three cases:
Speaking casually to a friend about a 先生 or other respected figure, as virmaior pointed out
In older materials, or materials set in older settings like 時代劇
In written materials like newspapers
The common thread here is a difference between the two types of polite language. 敬語 is about reinforcing the social order: respecting the acknowledged hierarchy that exists between samurai and lords, or between shopkeepers and customers, regardless of their personal relationship or attitudes. 丁寧語 is more about the relationship between speaker and interlocutor: being polite to someone as a sign of personal respect, unfamiliarity, and/or distance.
So getting back to the specifics, the lack of です・ます体 in newspapers is probably mostly about saving space, but it only flies because there's not as direct of a connection between the writer and reader, which is also why you don't usually see it in, say, novels - the point is to convey information or a story, not create a strong relationship with the reader. It has more of a place in spoken language. And the type of 謙譲語有り・丁寧語無し speech that sazarando pointed out makes sense for a samurai who knows his place but also isn't going out of his way to ingratiate himself. (It's also not that unusual for the time; somewhere along the way people went from using 普通形 by default unless they wanted to be polite to using 丁寧語 by default unless they're familiar.) For customer service or business situations (which is where you mostly hear 敬語 nowadays), you can see why you'd want to use both to emphasize both respect and politeness.
In earlier time, say pre- or mid-war Japan, people were more conscious of whom, when and where they should use 丁寧語, 謙譲語, 尊敬語 and distinction of their usages. But it seems people are not so serious today in observing the rules of the verbal locutions of expressing politeness, humiliation and respect.
In fact, there is a prevailing phenomenon of “ため口をきく” meaning a junior addresses and speak to a senior in the manner of equal position and status among young generations.
Today, the strict rules and manners of 丁寧語, 謙譲語, 尊敬語 are observed only in places such as among the shop clerks in department stores, receptionists and service staffs in high-class hotels, restaurants, and air-counters, where they are traind to observe the rules in speaking to customers.
With that said, all the examples of 参る, 申す, いたす, you quoted, except 戴く(頂く）sound old-fashioned, or are almost obsolete today, at least as a spoken language. However, 戴く(頂く）is frequently used today as in:
先ほどお電話を頂きましたが - I received a phone from you a few minute ago,…
これ、頂いてもいいですか – Can I have this one?
We used to say and write:
1.明日１０時に参上いたします - I’ll come to you at 10:00 am.
2.そういう具合には参らない – It won’t go like that.
3.そのように申し上げましたが – I said so, but …
4.私、田中と申します – My name is Tanaka.
5.今後一層努力いたす所存でございます – We will make further efforts (to comply with your expectation).
But all of these expressions except 4. appear antiquated today, and only are surviving in 時代劇 - period dramas, like Shakespean dramas.