"elements required by agent" is "required" here an adjective or verb?

It could be either a "past participle"/passive participle or an adjective

Most grammatical analyses do recognize a distinction between two types of -en/-ed words: one category comprises words that are clearly adjectives, while the other category is a bit harder to analyze, but is often analyzed as verbs that are "used as adjectives". The term "participle" may be reserved for this second category (of "verbs used as adjectives"), and in that case, the words in the first category are called something like "participial adjectives" or "departicipial adjectives".

Unfortunately, it's not always possible to determine which category a word belongs to. There are various tests, but most of them only apply to specific contexts. For example, if a word is modified by a directly preceding "very"--as in "a very excited child"--it must be a departicipial adjective, because "very" by itself can't modify a verb.

One rule that I think may be relevant for the analysis of "required" is un-prefixation. The negative prefix un- typically attaches to adjectives, not verbs (there is a separate verbal prefix un-, but it primarily has a "reversative" rather than a "negative" meaning--you can contrast the way the two prefixes work by considering the two possible meanings of "undone": "not done" and "reversed").

It seems to me that "required by" can be negated by adding un- to it: "elements unrequired by [agent]". I would say that this is evidence that "required by" can be an adjective. But it doesn't necessarily show that "required by" is always an adjective.

The "un-" test also indicates that "expressed by" can be an adjective.

The presence of a phrase starting with "by" is sometimes considered to be an indicator that a word is a participle rather than a departicipial adjective, but I have found this to be an unreliable rule, because its results are often incompatible with the results of the un- test or of the very test. See my answer to this related Linguistics question for more details about that: verbal or adjectival suffix -ed in the word “excited”

"Adjectival participles revisited", by Artemis Alexiadou, Berit Gehrke and Florian Schäfer (2012), gives the following data and summary:

3.1 By-phrases and event-related modification in adjectival passives

• The standard claim for English is that adjectival passives are incompatible with by-phrases, and to our knowledge there is little discussion on other event-related modifiers in English adjectival passives.

• The generality of this claim has been challenged recently for many languages, for which it has been shown that event-related modifiers are available [...]

English data from McIntyre (2011) and Bruening (2012):

(19) a. The dictator remained unsupported/underestimated by the warlords.
b. Former investigator says he remains disturbed by what he saw at baby murder scene.
c. No longer does Tim Thomas appear trained by Tim Hortons.
d. There are others who I would call saints more than theologians since they seem taught by God more than by men.
e. Once one monkey discovered a new food-washing method, very soon the whole tribe used the method, untaught by the original simian.
f. Steve Jobs’ birthday doesn’t go unnoticed by spammers.

➩ There exist adjectival passives with by-phrases also in English and German.
➩ But these are restricted compared to by-phrases in the verbal passive, or compared to by-phrases in Greek adjectival passives.

(pp. 6-7)