No.
Character Advancement
When adding new levels of an existing class or adding levels of a new
class (see Multiclassing, below), make sure to take the following
steps in order. First, select your new class level. You must be able
to qualify for this level before any of the following adjustments are
made. Second, apply any ability score increases due to gaining a
level. Third, integrate all of the level's class abilities and then
roll for additional hit points. Finally, add new skills and feats.
The order of how you level matters. You're allowed take a feat the same level you gain the prerequisites only because gaining feats is the last thing you do. If the prerequisite is a skill, spell, or ability, you already have it by the time you take the feat.
Of course, if the prerequisite is another feat, you must already have the prerequisite feat before you can take the desired feat. However, since character advancement is an ordered process, if you gain multiple feats in a level1, you can choose to take the prerequisite feat first. You can now take the desired feat.
In your example, you can take neither feat first since neither feat has its prerequisite met.
1 Class abilities that give bonus feats give them at the time you normally add new feats (after skill points). It's unclear if you choose your standard feat before your bonus feat or vice versa, but I would generally allow players to select feats in any order if it matters.
Yes, they would for the skills that are class skills (Acrobatics, Climb, Fly, Perception, Stealth, and Swim).
http://paizo.com/pathfinderRPG/prd/usingSkills.html
Each skill rank grants a +1 bonus on checks made using that skill.
When you make a skill check, you roll 1d20 and then add your ranks and
the appropriate ability score modifier to the result of this check. If
the skill you're using is a class skill (and you have invested ranks
into that skill), you gain a +3 bonus on the check. If you are not
trained in the skill (and if the skill may be used untrained), you may
still attempt the skill, but you use only the bonus (or penalty)
provided by the associated ability score modifier to modify the check.
Skills can be further modified by a wide variety of sources—by your
race, by a class ability, by equipment, by spell effects or magic
items, and so on. See Table: Skill Check Bonuses for a summary of
skill check bonuses.
Best Answer
As the other answer points out correctly - generally feats are only gained when you gain hit dice.
However arguably the Cavalier ability Tactician could grant Teamwork feats to a mount, though a GM might rule against it.
More usefully however the feat Animal Ally would allow you to gain an Animal Companion, which will gain hit dice as you level and thus gain feats (note that many feats and skills require an Int of 3* or greater, but the animal companion will gain stat bonuses every 4 levels, just as a character does). http://paizo.com/threads/rzs2ktcl?Animal-Companion-with-3-INT
The Monstrous Mounts section of the leadership feat may also be worth a look.
Familiars generally don't gain feats and generally can't be used as mounts, but they do gain skills. There are exceptions, such as a Beast-Bonded Witch, and feats you can take to improve or evolve familiars. There are lots of different ways to gain familiars - there may be one available to you.