Is association, conditioning, and symbolic learning the same thing?
Conditioning (both classical and operant), memory, and others mentioned in the question are considered examples (types) of association by associationism, a school of philosophy in psychology that suggests that all mental processes may be based on similar or proximal mental states. Usually this idea is too broad and vague to be very practical to apply, but it has spawned a number of useful fields of study, including connectionism, a general strategy used in neural networks and learning modelling, and schemata, used in memory research.
Is associating things an entirely different process from linking
things together to create symbols?
Using connectionist models, artificial neural networks have demonstrated learning in a variety of subject areas, including operant and classical conditioning (examples of association), and handwriting recognition and sign reading (examples of symbols). This suggests that the process involved may be essentially the same.
One of the major challenges to basic associationism has been that in practice, associations do not seem to obey any straightforward rules. For example, children learn to associate words like "food" with their meaning very quickly with few (sometimes no) repetitions, while branding requires many more repetitions, and other associations (there is nothing intrinsically bad about this song) may never be learned. For this reason, other, more rigorous theories, such as conditioning, are preferred.