Learn Russian: Всякий гибкий человек может приспосабливаться к любым условиям. - Every sort of flexible person can adapt to absolutely any conditions.

This highlights the exact difference between them! Всякий refers to every person of a certain type (here, the flexible type). Любой means absolutely any, no matter which. If you said Любой гибкий человек, it sounds a bit unnatural, like saying take your pick of flexible people. Using всякий categorizes the person, while любым shows there is zero restriction on the conditions.

Why is условиям in the dative case here?

It is triggered by the preposition к. The reflexive verb приспосабливаться (to adapt) pairs with к plus the dative case to express what you are adapting to. It gives the feeling of adjusting yourself toward something.

Why use the imperfective verb приспосабливаться instead of the perfective приспособиться?

The sentence describes a general characteristic rather than a single, specific event that will definitely be completed. When stating a general truth or an open-ended ability (what a flexible person can do in general), Russian uses the imperfective aspect.

Does the word гибкий normally mean physically flexible, or is it usually metaphorical like in English?

Just like in English, гибкий means both. You can use it literally for a bendy tree branch or a physically flexible athlete, and metaphorically for an adaptable mindset, a flexible schedule, or flexible prices.