Does every online algorithm has an offline counterpart?
This statement seems false to be because of the notion of competitive ratio that is used to analyze virtually all online algorithms. Heuristically the competitive ratio measures how well the online algorithm performs on a sequence of inputs versus an adversary that knows the sequence in advance. In particular, the adversary is solving an offline problem. 

If you look at any of the online problems that are studied, they all use the competitive ratio as a benchmark. To back up this claim, look at the abstracts of any paper in a well known journal (such as SODA 18: https://archive.siam.org/meetings/da18/da18_abstracts.pdf) and search for abstracts that study online algorithms. They all use some type of competitive ratio in their analysis.