Can I increase the number of trained skills if I increase the Intelligence modifier

Your first option is correct - you choose from the options listed in the "Key Ability" list of the class in question. Rogue is slightly different because its Key Ability is listed as "Dexterity or Other", where "Other" is clarified to mean "an option from rogue's racket", and since you do not get a racket, you have no "option from rogue's racket" to choose, requiring you to choose Dex.

There isn't any specific rule I can find that states this unequivocally, but per the quote you include concerning Class DCs, it is equal to 10 plus proficiency (2+level for Trained) plus the modifier for the class's key ability score, which is the exact words used in the class features list I have called out above, as well as clarified in the description for Key Ability, so I have a high level of confidence in this interpretation.

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.