Learn Finnish: Meillä ei ole ollenkaan kofeiinia. - We do not have any caffeine at all.

Yes, you can replace ollenkaan with lainkaan without changing the meaning. Both mean "at all" in negative sentences. The only difference is that lainkaan sounds slightly more formal or literary, while ollenkaan is very common in everyday spoken Finnish.

Why does ollenkaan come before kofeiinia, when in English "at all" goes at the end of the sentence?

In Finnish, modifying adverbs like ollenkaan usually sit right after the verb they modify—in this case, the negative verb block ei ole. You could actually say Meillä ei ole kofeiinia ollenkaan, but putting it right after the verb is the most neutral and natural word order.

As a refresher, why must kofeiinia be in the partitive case here?

Two rules overlap to cause this. First, this is a negative "having" sentence (ei ole), and the object of a negative possession structure must always be in the partitive. Second, caffeine is an uncountable mass noun, which often takes the partitive even in positive sentences when referring to an unspecified amount.

Does the word ollenkaan originate from the verb olla (to be)?

It does! It is a historically frozen form of the verb olla combined with the suffix -kaan (which adds emphasis or means "either" in negative contexts). You don't need to worry about conjugating it; it is always used as a fixed adverb.