Actually monstrous mount and monstrous mount mastery are only necessary if you want to hook the animal companion abilities and progression onto the griffon. If you are ok with it just having 'regular griffon stats' then all you need is handle animal skill or diplomacy.
Depending on what you want from the griffon and cohort this may be just fine. An archer character where the mount is just to keep them up in the air and away from trouble will probably be ok with a 5HD mount well into the mid teens.
That opens up quite a few options, possibly even having the griffon just be the mount for your wizard character.
That is true to an extent. But also remember at level 12 your cohort will be level 10, his animal companion will by 9HD. Considerably more than a regular 5 HD griffon, but if it is also expected to be in the thick of things, rather than a mount designed to keep you at a distance it will probably be less survivable. It depends on your GM to an extent of course, but usually when I run games intelligent foes will concentrate on taking down bigger threats rather than smaller ones and so I'd generally be having the big bad try to take your character down, not your mount since succeeding on that will be an inconvenience, but hardly critical to your success at combat. And of course as a wizard you do have some options to increase it's ability to survive.
Overland flight is pretty good, but it also means you are spending 5th level spell for a lower flight speed and less options, a mounted wizard can move and cast a full round spell for example.
I tend to think that two feats is way too expensive for what you get, and I think their are a lot better options, especially when looking at lower cohort levels anyway.
Depends a lot on what the two of you want to do. Most inquisitors I have seen go for a striker role, and wizards are great battlefield controllers. Given that, ideally you would want a good support character and a striker.
Personally, I like bards or evangelist clerics best for support. For you guys, the inquisitor can deal with a lot of the condition removal if necessary (particularly if you buy scrolls) and do out of combat healing so the difference is really preference. I'd probably lean toward a bard, maybe a small race with the duettist archetype and a flying mauler familiar so their would be a couple of airborne people. First round haste and inspire courage (once cohort level 8) is pretty impressive first round support. After that, other than some emergency actions I'd just have him mostly shoot some arrows.
For a striker (since it sounds like you both getting cohorts) I'd be looking at pretty much any full bab. The 3/4 babs can do the job too, and I prefer them when playing, but they are typically a lot more complex to run, and for a cohort I'd prefer something simpler. Since you were mentioning it before, Paladin would be a pretty strong choice, giving you a little more emergency healing (and healing that is already in the front line) and would thematically fit well with an inquisitor character.