So let me deal with each question separately:
FIrst off, make sure you do not confuse adjectives and adverbs. Apples are more healthy to eat than caramels.
Here we are comparing two nouns. The word 'healthy' modifies those nouns and is therefore an adjective. Now take a look at the following sentence:Tim eats more healthily than Mike does.
We are not comparing Tim and Mike, but the way in which they eat. How somebody "verbs" is an adverb. In this case, healthily describes the verb eat and is therefore an adjective. Using this same logic, now let's look at the sentence you originally provided. Adrian runs quicker than Jacob.
Here we are comparing how they run. Therefore we want an adverb (quickly) not an adjective (quicker). Because we removed the -er ending, we no longer have a comparison. Therefore we need to add the 'more' next to the adverb 'quickly' so the sentence should read:Adrian runs more quickly than Jacob.
With that in mind, which word is correct in the following sentence:Adrian is quicker/more quickly than Jacob.
Because we are comparing two nouns (Adrian and Jacob), 'quicker' is correct.
Now to @Shikhar's question. When comparing how two different nouns verb, as in Jacob runs faster than Adrian, the 'does' is optional. In English, this is called an ellipsis, or an omission of a word, when the meaning is clear from the context. Here it is clear that we are comparing how fast the two run. Therefore, the 'does', while not incorrect, is consider superfluous.
Hope that helps
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