Picking one archetype grants that one archetype's features; picking more than one compatible archetype grants all archetypes' features

Archetypes are picked when a character first takes a level in the class and persist throughout the character's career, forever altering that class for that character (barring, for example, retraining).

When a character picks an archetype, he gains that archetype's special abilities whenever he gains a level in that class. If the character picked for a class more than one compatible archetype (i.e. the two or more picked archetypes lack features that each replaces), he gains the special abilities of all picked archetypes for that class.

The section Archetypes on Alternate Class Feature says that

When an archetype includes multiple alternate class features, a character must take them all—often blocking the character from ever gaining certain standard class features, but replacing them with other options. All other class features of the base class that aren't mentioned among the alternate class features remain unchanged and are acquired normally when the character reaches the appropriate level, unless noted otherwise. A character who takes an alternate class feature does not count as having the class feature that was replaced for the purposes of meeting any requirements or prerequisites.

A character can take more than one archetype and garner additional alternate class features, but none of the alternate class features can replace or alter the same class feature from the base class as another alternate class feature. For example, a fighter could not be both an armor master and a brawler, since both archetypes replace the weapon training 1 class feature with something different.

Emphasis mine. When combining archetypes, there's no other option but to use these rules. There's no way to take, for example, levels in skulking slayer rogue then levels in scout rogue; instead, a creature takes levels in scout skulking slayer rogue, getting each archetype's benefits and drawbacks.

This situation in particular

Hence, on becoming a level 4 rogue, a half-orc rogue who had picked the archetypes scout and skulking slayer would have the following from the skulking slayer archetype:

• The altered class skills and adjustment to skill ranks.
• The altered weapon and armor proficiency.
• The special ability pass for human.
• The special ability underhanded maneuvers.
• The special ability bonus feats, allowing the selection of the Surprise Follow-Through feat in place of a rogue talent.
• The special ability bold strike.

And would also have the following from the scout archetype:

• The special ability scout’s charge.

At each level of rogue such a character doesn't pick which archetype applies; instead, each archetype applies automatically.

From a strict reading, the mount does not have Share Spells (even though it's an animal companion, and they usually do) because it's a Beast Rider mount (different from normal Animal Companions). Note here that it does not say that it loses the Share Spells ability, merely that it does not gain it. However, it is also a Cavalier mount, and therefore should be treated as having Share Spells for the purpose of qualifying for archetypes. (This is a specific exception, as Cavalier mounts usually do not have that ability. This mount also does not have that ability.) There does not appear to be a conflict here.

By extension, RAW it should be able to qualify for animal archetypes (and only animal archetypes) as if it had Share Spells.

Technically, if you really wanted to cheese it for RAW, you could argue that it would continue to qualify as such even if you had taken another archetype that also removed Share Spells, but that, I think, would be pushing things.