[RPG] ny consequence after Fatigued if you continue going without rest

If you're a mythic character, then Recuperation gives you a full night's rest in 1 hour, including refreshing daily abilities such as spells. You can do this multiple times per day, limited only by your mythic power points. The item Cornucopia of Plenty makes this more power-efficient, but still requires a mythic character.

This is a rules blind-spot, but context indicates that animal companions should have broad access to skills.

You have highlighted the specific rules hiccups that occur if you assume animal companions are limited to the actions in the Command an Animal action. You have also highlighted the specific animal companion abilities that, from context, appear to assume that most animal companions are capable of using a broad range of actions, including Sneak, Fly and Hide.

Note however that Animal Companions are never explicitly limited to the actions described under Command an Animal. Bolding mine below.

Most animals know the Drop Prone, Leap, Seek, Stand, Stride, and Strike basic actions.

Animal Companions are not "most animals". They have exceptional/unusual statistics and unique capabilities in comparison to their non-Companion equivalents. Their statistics grow in pace with their master and by mid or high levels vastly eclipse the animal they are based on.

The only explicit limit on skills/actions comes under the Proficiencies section (which you've highlighted in your question). From Animal Companion Proficiencies:

Animal companions can't use abilities that require greater Intelligence, such as Coerce or Decipher Writing, even if trained in the appropriate skill, unless they have a specialization that allows it.

This suggests that Animal Companions are allowed to use all generally available actions and abilities that don't require "greater Intelligence", as interpreted by the GM. To the best of my knowledge and research the system's developers have not commented further on how the GM should interpret which actions take more or less intelligence. As a result, this falls to GM discretion.