What are the repeatedly used structures in 4-character expressions (e.g. chengyu) and what do they mean?

Axel Tong answered A而B之 perfectly. I would address some other idiom structures not mentioned in the question

Becky 李蓓 wrote:

Note that x而y之 is just one example, and there are others like 千x万y

1. [(adjective A) + (noun A) + (adjective B) + (noun B)] / Example: 千丝万缕，如狼似虎

2. [(adverb A) + (adjective A) + (adverb B) + (adjective B)] / Example: 穷凶极恶，巨奸大恶

3. [adverb A] + [(verb A) + (adverb B) + (verb B)] / Example: 穷追猛打，虚打假闘

4. [(verb A) + (adverb A) + (verb B) + (adverb B)] / Example: 赶尽杀绝，看惯见熟

In the four structures above, the relationship between (A) and (B) are either being similar or being one half of a split-up compound word

千万 丝缕 --> 千丝万缕

如似 狼虎 --> 如狼似虎

穷极 凶恶 --> 穷凶极恶

巨大 奸恶 --> 巨奸大恶

穷猛 追打 --> 穷追猛打

虚假 打闘 --> 虚打假闘

赶杀 尽绝 --> 赶尽杀绝

看见 惯熟 --> 看惯见熟

Studying the examples about, you can apply the same structure to coin new idiomatic phrases out of the common compound words that contain two characters that have a similar, identical or opposite meaning

Example:

粉碎 身(体) 骨(骼) --> 粉身碎骨

超赶 英美 --> 超英赶美

忽然 冷热 --> 忽冷忽热

It seems that there are two types of 成语 represented by the same “structure”.

A而B之 could mean “action + action, done to something (third person pronoun)”.

取而代之 means “to take and replace 之 (something that was previously mentioned)”. Compare this to 取代, which simply means “replace”. You could consider 取而代之 a literary form of 取代; using it adds elegance to an otherwise common word. Other examples include 换而言之、敬而远之 and 分而治之.

A而B之 could also mean “adjective + adjective, used to describe something (third person pronoun)”.

堂而皇之 means that something (之) is “wide like a hall and grand like imperial edifices”. Similarly, this can be considered a literary form of 堂皇. I would say 久而久之 has similar characteristics, but note that the 久 is describing a long period of time, instead of the appearance or a quality of an object.