Can a Cavalier choose Eagle Mount at first level?

Question

A level one Cavalier receives a mount as a special class feature.

Can cavailers get riding eagles or hawks at first level?

Riding Eagle

Special: A cavalier can choose this as a mount.

Size Large
Speed 10 ft., fly 80 ft. (average)
AC +0 natural armor
Attack bite (1d6), 2 talons (1d4)

Ability Scores:
Str 14
Dex 13
Con 12
Int 2
Wis 12
Cha 8

Special Qualities low-light vision.

Riding Hawk

Special: A cavalier can choose this as a mount.

Size Medium
Speed 10 ft., fly 60 ft. (average)
AC +1 natural armor
Attack bite (1d6), 2 talons (1d4)

Ability Scores:
Str 12
Dex 13
Con 14
Int 2
Wis 12
Cha 6

Special Qualities low-light vision.

Answer 1

Yes, any animal is available if your GM allows it

The core of the Mount class ability is that any mount your GM feels that is suitable to your character can be used as a mount, as said on it's description:

Mount (Ex)

A cavalier gains the service of a loyal and trusty steed to carry him into battle. This mount functions as a druid’s animal companion, using the cavalier’s level as his effective druid level. The creature must be one that he is capable of riding and is suitable as a mount.

A Medium cavalier can select a camel or a horse. A Small cavalier can select a pony or wolf, but can also select a boar or a dog if he is at least 4th level. The GM might approve other animals as suitable mounts.

Normally, this mount is a horse-like animal, four-legged and usually one size category larger. But the class ability does not restrict your character in no way, and the GM is free to allow any mount that would make sense for him.

That mount is not restricted to only one of those listed as examples, unlike the Ranger's animal companion, for example, that has a restricted list (even so, I know a lot of GMs don't enforce it based on community feedback):

A ranger who selects an animal companion can choose from the following list: badger, bird, camel, cat (small), dire rat, dog, horse, pony, snake (viper or constrictor), or wolf.

That said, we can only tell you that GM's normally approve those listed on the Mount class ability, but GM's should keep in mind that the ability is giving them extra power to change the available mounts to cavalier characters without actually house ruling to favor your character.

For an official source of this, on the Players Guide of each Adventure Path, there are tips and hints about animal companions and mounts available specifically for that adventure, based on the geographical location of the adventure, usually restricting the options available. But sometimes also expanding the options for druids, rangers, cavaliers and hunters.

For example, the Iron God's Players Guide (free download) says the following:

Animal Companions and Familiars

The following animal companions can all be found in Numeria: antelope, aurochs, badger, bear, bird, boar, dire bat, dire rat, dog, elk, giant weasel, horse, mastodon, moose, pony, ram, stag, viper, and wolf.

The following familiars are commonly found serving wizards and witches in Numeria: armadillo, bat, cat, centipede, fox, goat, giant f lea, hawk, hedgehog, owl, pig, raccoon, rat, raven, scorpion, skunk, squirrel, f lying squirrel, toad, viper, and weasel.

In addition to these suggestions, three new alien animals are presented in the bestiary of the first volume of the Iron Gods Adventure Path. Talk to your GM about these animals and see if they are a good fit for your character and campaign.

Here, the three animals (petromin, pilo and sorico) are alien from Golarion and actually come from another planet, but are available to be selected for characters playing Iron Gods (set in Numeria).

Riding Eagles are a bad choice for Medium-sized characters

Note that even lightly armored characters might make this impossible due to weight and the carrying capacity of the (3PP) riding eagle (116 lbs. light load). Even the lightest medium characters on the core rulebook will weight at least 95 lbs (female human, female elves closely behind with 96 lbs), add a bare minimum of equipment and the mount will already be on a medium load (-3 on fly checks).

\begin{array}{|c|c|c|}\hline \mbox{Strength Score}& \mbox{Light Load} & \mbox{Medium Load} & \mbox{Heavy Load} \\\hline \mbox{14} & \mbox{116 lbs. or less} & \mbox{117–232lbs.} & \mbox{232–350 lbs.} \\\hline \end{array}

With average fly speed (+0 on checks) large size (-2 on checks) and dex 13 (+1 on checks), you are looking at an excuse to plummet to the ground in combat. I would advise against it. The best possible check at first level is a bonus of +3 (+1 dex modifier, 1 skill rank at 1st level, +3 class skill, +3 skill focus, -2 penalty from size, -3 penalty from weight), which is not enough to automatically pass on DC 10 checks in stressful situations. Any extra equipment, such as the character also carrying a medium load, will only make this harder on your mount (-6 penalty on heavy load).

A Roc animal companion is a much better choice in the long run (7th and up), though they also start with a very low str score (str 12) and being medium-sized creatures (large at 7th with +8 strength). They will have about 22 strength at 7th level (+2 from class advancements), which is enough to carry a medium-sized cavalier with some armor, weapons and his equipment, as long as he doesn't exaggerate. But you will not be riding this mount at the earlier levels (1-6) without the help of magic.

\begin{array}{|c|c|c|}\hline \mbox{Strength Score}& \mbox{Light Load} & \mbox{Medium Load} & \mbox{Heavy Load} \\\hline \mbox{22} & \mbox{346 lbs. or less} & \mbox{347–692 lbs.} & \mbox{693–1040 lbs.} \\\hline \end{array}

Answer 2

At level 1 a Medium cavalier can pick the riding eagle and a Small cavalier can pick the riding hawk

To serve as the cavalier's mount, "[t]he creature must be one that [the cavalier] is capable of riding and is suitable as a mount." Normally, a mount must be at least one size category than the rider, so the riding hawk (and pony and wolf and boar and dog) can't typically be a Medium cavalier's mount. However, the player of a Medium cavalier can ask the GM if Medium mounts are available to his PC if, for instance, the PC possesses the feat Undersized Mount.

There is, so far as I'm aware, no rule like The mount must be one size category larger than the Cavalier until the cavalier is level 4 when it may be the same size category. The cavalier's mounts are organized by the size category of the cavalier: "A Medium cavalier can select a camel or a horse" (both Large mounts), and, "A Small cavalier can select a pony or wolf, but can also select a boar or a dog if he is at least 4th level" (all Medium mounts, the boar and dog likely being considered more powerful—hence the delay in acquisition—than the pony and wolf).

And while "[t]he GM might approve other animals as suitable mounts," the riding eagle and riding hawk can fly, which, at level 1, is a red flag for many GMs, despite the former's carrying capacity of 116 lbs. and the latter's of 43 lbs. (making for some skinny and lightly armored on-the-wing cavaliers!). So while these mounts may be possible if the GM's given blanket approval for third-party material, the GM may still say No, usually because flying at low levels obviates many typical threats. (Also see this question.)