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.
Spell Casting Is Prevented
The Polymorph trait states that:
If you take on a battle form with a polymorph spell, the special statistics can be adjusted only by circumstance bonuses, status bonuses, and penalties. Unless otherwise noted, the battle form prevents you from casting spells, speaking, and using most manipulate actions that require hands. (If there's doubt about whether you can use an action, the GM decides.)
There is no ambiguity1, battle forms prevent you from casting spells of all kinds, including spells that don't require hands.
As you've pointed out, you can benefit from ongoing spells that you cast beforehand, including using actions granted by such spells... as long as said benefit doesn't conflict with the quote above. For example, you can't benefit from Mage Armor's item bonus, and without GM fiat you can't snap your fingers as part of the manipulate action granted by Instant Armor, but problem spells like these are few and far between.
- To see why, take a closer look:
the battle form prevents you from casting spells, speaking, and using most manipulate actions that require hands.
In this sentence, 'that require hands' must refer to either the entire list 'casting spells, speaking, and using most manipulate actions' or just 'manipulate actions'.
However, the syntax just doesn't work for the entire list and this becomes especially obvious if we rewrite the sentence in full for each item in the list:
the battle form prevents you from casting spells that require hands, it prevents you from speaking that require hands, and it prevents you from using most manipulate actions that require hands.
Hence, 'that require hands' must refer only to 'manipulate actions'.
Best Answer
Many, if not most, property runes don't work
Polymorph states:
Runes that function through activations like earthbinding and extending don't work, because you can't activate them.
For runes that add extra damage to attacks we can get a solid clue elsewhere. Namely, dragon transformation, which states:
This feat basically confirms that extra damage is an adjustment of the form's statistics, or else you'd be able to add your rage damage to the dragon form even without it. Property runes like shock or disrupting which increase your damage likewise do so via extra damage, not via circumstance or status bonuses, so their damage does not work.
And as you've already pointed out, lines like "plus 1d6 fire" that are found in some battle forms also support this interpretation.
The fate of the other property runes is more ambigous. The notion that "the special statistics can be adjusted only by circumstance bonuses, status bonuses, and penalties" could be reasonably be refering to numerical statistics only, because all the adjustments that do work are numerical too. In this interpretation the rest of the property runes by and large work with the battle forms.
However, a GM could reasonably rule that adjustments to any kind of statistic must be circumstance or status bonuses. The statistics of battle forms are clearly similar in format to creature statistics, especially the attacks, so it wouldn't be a stretch to read the form's attacks through the lens of Reading Creature Statistics:
Here we can see that everything from the traits to the additional effects are part of a creature's attack statistic, and this is probably true for battle form statistics too. Just like with extra damage, we can find circumstancial evidence for this in some battle forms. For example, some of the aberrant form's attacks say "and you can spend an action after a hit to Grab the target", one of the choral angel form's attack says "deafened for 1 round on a critical hit", the moon cosmic form says "on a critical hit with either Strike, the target is stupefied 2 for 1 round", etc.
So if the GM rules that non-numerical statistics must also be adjusted by circumstance or status bonuses, then property runes which change weapon traits like merciful and hooked, and property runes which grant additional effects to attacks such as ghost touch and (the on crit effects of) distrupting do not work for battle forms.
In short, activations and extra damage don't work. Trait changes and additional effects are up to GM interpretation. I believe those 4 categoriers catch almost every property rune, with these few exceptions.