Beastrider
I'd suggest changing the Animal Bond bonus to occur at every even numbered level only, dropping the bonus at 1st. This leads to a 20th level Beastrider getting a +10 bonus. That appeals to my sense of neatness
A much more significant change that I would suggest is to change the Mounted Feat progression from its current run (1st, 5th, 9th, 13th and 17th) as this leads to the strange situation that a Fighter could basically have all five Mounted Feats by 4th level (if they took nothing else), while a Beastrider must wait until 17th. Beastriders will, I think, want the Mounted Feats much earlier than that. I suggest giving the feats at 1st, 3rd, 5th, 7th and 9th.
It's not specified in the write-up, but I think you should make it clear that if a Beastrider's mount dies, then it takes a year for their next mount to develop any Advancement abilities the last mount may have had. Basically the same thing as a Paladin, to emphasis the importance of the link between mount and rider.
Everything else seemed fine, except that I am not sure why Concentration is a class skill, and I think Listen should probably be on the list of class skills.
Archer
I'm not a fan of the "two BABs" idea, so I'd suggest changing the class to be "BAB as Rogue", with a special class ability:
Ranged Combatant: the Archer gains +1 to hit with all ranged attacks. This bonus stacks with all other bonuses. It increases to +2 at 5th, +3 at 9th, +4 at 13th and +5 at 17th levels.
This means the Archer has only one BAB, but comes close to mimicking the higher BAB when using ranged weapons. The difference is that they don't get extra iterative attacks from this ability, which they do from an increase in BAB.
I'd also suggest changing "Archery" to simply Free Feats, full strength, without the light armour restriction, as the restriction is pretty meaningless for a high Dex character, anyway. I'd provide the feats at 2nd, 5th, 8th, 11th, 14th, 17th and 20th levels. The list of feats would be the same as you have given, with Weapo Specialisation (ranged weapons only) also added to the list, and the "any bow" feats expanded to "any ranged weapon".
I'd drop the Bow Specialisation currently given at 5th level, as well as the Deadly Aim (2nd) and Defensive Firing (8th).
Point Blank Shot would remain as a free feat at 1st level.
Finally, I'd change the Special Abilities to be given at 7th, 13th and 19th levels and change the list as follows:
Add Deadly Aim and Defensive Firing as special abilities. Remove the Concentration Checks involved in both skills. Limit the Deadly Aim to the same as the "Ranged Combatant" bonus above, with each +1 taking a full round action to gain. Change the "no target movement" to "no target movement, or target is unaware of the archer's presence".
Make Defensive Firing allow the Archer to use missiles of any kind without an AoO, no Concentration check required. It's not that big a deal.
Remove Armoured Archer - it is no longer required under this model.
Remove Bow Mastery - it's a bit too twinky
Critical Shot - increase the pre-req to "has improved critical with the weapon"
Power Shot - I'd change the name and explain the bonus as targetting particularly vital parts of the enemy, rather than the "power" of the shot, but the mechanics are fine.
Quickstring - change the ability so that, instead of the current concept, it changes the Archer's BAB with Ranged Weapons to mimic the Unarmed BAB of a Monk. ie iterative attacks occur every -3, not every -5. Allow other BABs to stack with this ability (since it won't be available until 7th, anyway)
Sniper Attack - make this 1d6 per time it is taken, rather than 2d6 the first time, then 1d6 after that. 2d6 in one hit is too much.
Storm of Arrows - this is too powerful, IMO, and should be removed.
Trick Shot - this ability overlaps the Sharp-shooting feat, and woulfd be problematic for adjudication. I'd say it would be ebst to remove it, and just use the Sharp-shooting feat.
For the class skills list, I'd remove Concentration and add in MOve Silently and Handle Animal. This fits better with the ranger/scout kind of idea.
Awareness Barbarian
I don't get the -2 penalty to damage? What is the reason for this? If the intent is just to balance the bonuses given by the ability, I'd suggest just dropping the bonus to hit, and this penalty, and leaving it at +4 Dex and +1 morale bonus on all checks. The improved version would be +6 Dex and +2 to checks.
Also, thinking about it, Scout is a poor choice of name for this class: it suggests sneaking and spotting skills that don't really fit into the class's make-up. Instead, how about "Dervish"?
