Are monks expected to accept life lessons from (time spent with) the laity?

I think there were several cases where the Buddha "learned life lessons" from monks's interactions with laypeople -- this is one example.

So far as I know many of the Vinaya rules have an "origin story" in the commentary which explain why/when the rule was enacted.

Also the Udayi Sutta: About Udayin (AN 5.159) says,

"It's not easy to teach the Dhamma to others, Ananda. The Dhamma should be taught to others only when five qualities are established within the person teaching. Which five?

"[1] The Dhamma should be taught with the thought, 'I will speak step-by-step.'

"[2] The Dhamma should be taught with the thought, 'I will speak explaining the sequence [of cause & effect].'

"[3] The Dhamma should be taught with the thought, 'I will speak out of compassion.'

"[4] The Dhamma should be taught with the thought, 'I will speak not for the purpose of material reward.'

"[5] The Dhamma should be taught with the thought, 'I will speak without hurting myself or others.'[1]

... where, according to the footnote ...

According to the Commentary, "hurting oneself" means exalting oneself. "Hurting others" means putting other people down.

Perhaps that -- i.e. "exalting oneself and hurting others" -- is applicable to the question i.e. "the attitude they know better".

I expect that probably a monk does indeed or in fact know better -- so if it were "conceit" ...

Māna (Sanskrit, Pali; Tibetan: nga rgyal) is a Buddhist term that may be translated as "pride", "arrogance", or "conceit". It is defined as an inflated mind that makes whatever is suitable, such as wealth or learning, to be the foundation of pride. It creates the basis for disrespecting others and for the occurrence of suffering.

... then I guess it's likely to be "true conceit" as defined here -- i.e. "thinking one is superior when one is superior", if the venerable compares his attainments with laypeople's and vice versa -- which I guess is part of what might make that so difficult.

I'd also like to answer that no, I'm not aware of such a doctrine in the suttas. I gather that generally monks teach if they're invited to and only when the audience shows respect -- but it's difficult to prove this answer, i.e. difficult to "prove a negative".

Your example could be a case of conceit (mana), and is one of the ten fetters.

From Samyutta Nikaya, Sona sutta:

When any ascetics and brahmins, on the basis of feeling … on the basis of perception … on the basis of volitional formations … on the basis of consciousness—which is impermanent, suffering, and subject to change—regard themselves thus: ‘I am superior,’ or ‘I am equal,’ or ‘I am inferior,’ what is that due to apart from not seeing things as they really are?

/.../

Any kind of feeling whatsoever … Any kind of perception whatsoever … Any kind of volitional formations whatsoever … Any kind of consciousness whatsoever, whether past, future, or present, internal or external, gross or subtle, inferior or superior, far or near, all consciousness should be seen as it really is with correct wisdom thus: ‘This is not mine, this I am not, this is not my self.’

Seeing thus, Soṇa, the instructed noble disciple experiences revulsion towards form, revulsion towards feeling, revulsion towards perception, revulsion towards volitional formations, revulsion towards consciousness. Experiencing revulsion, he becomes dispassionate. Through dispassion his mind is liberated. When it is liberated there comes the knowledge: ‘It’s liberated.’ He understands: ‘Destroyed is birth, the holy life has been lived, what had to be done has been done, there is no more for this state of being.’”

https://suttacentral.net/sn22.49/en/bodhi

Yes, it has happened that the monastic Sangha learns from the lay Sangha. The vinaya has many cases where laity understood what was proper and complained about transgressions even before they became rules. For example, Venerable Udāyī was caught whispering in a lay woman's ear.

The training rule on teaching (origin story): How can Venerable Udāyī give teachings by whispering in the ear? Should not teachings be given audibly and openly?

This is from the origin story of the training rule on teaching.

In Ubon Ratchatani Luang Por Char talked about (translating from Isaan language) how "everything is teaching us."

Also, working with people was developing strength in him, he said.