When should I use 地 with an adjective and when should I not use it with an adverb (and why do some adverbs require 地)?
一个劲儿地 is not a standard adverb, It is an adverbial phrase
一个劲儿 (constant effort) is a noun. Adding 地 make it an adverbial phrase 一个劲儿地 (with constant effort)
刻苦 is not only an adjective, it can also be an adverb or noun

Is it common to omit 地 on an adjective, even if the adjective works as an adverb?


強烈反對 could mean 'strong objection [強烈 (adj) 反對 (n)] or 'strongly object' [強烈 (adv ) 反對 (v)] .

強烈地反對 could only mean 'strongly object'. Without another context, you cannot omit 地 when you meant to use it as an adverb


You can only omit 地 when the context clearly indicated 強烈 is an adverb, not an adjective. For example, 他強烈地反對 (he strongly objects) can be reduced to 他強烈反對

猛烈攻擊 could mean 'fierce (adj) attack (n)' or 'fiercely (adv) attack (v)'

猛烈地攻擊 could only mean 'fiercely attack'. Without another context, you cannot omit 地 when you meant to use it as an adverb


You can only omit 地 when the context clearly indicated 猛烈 is an adverb, not an adjective. For example: '對他猛烈地攻擊' (fiercely attack him) can be reduced to '對他猛烈攻擊'

Does it depends on each adjective that it is OK to omit 地? In other words, you can omit 地 on some adjectives but cannot on others?

See the examples above. Notice, an adjective that is exclusively for nouns cannot be turned into an adverb by adding 地. For example 強大，細小，好，壞

Is it common to use 地 on adverbs as well? If that is the case, why do adverbs need 地 in the first place?


Adverb with 地 emphasizes 'the manner' of the action. For example '大力地打'

Adverb without 地 focus on the verb itself, with the adverb as an equal-weighted complement. For example '大力打'

Single character adverbs usually don't use 地. For example '痛打'，'狠批' (never '痛地打'，'狠地批')



Likewise does it depend on each adverb if it requires 地?

Yes, usually adverb for verb can be used with or without 地. (see example above).  Adverb for adjective does not need 地, E.g. 局部成功,  絶頂聰明 (never 局部地成功, 絶頂地聰明)