1) I don't think so... Grapple is more of a "I'm holding your arm". although if you managed to Pin Down the flying creature, I think it'd fall.
2) I don't know the RAW answer, but I'd say yes. It just makes sense.
Somewhat of a side note... But I wish PF had kept the 3.5 rules for tripping flying creatures... -.-'
1) A flying creature cannot fly if it is carrying a medium or heavy load. I would count the monk's weight against this. The rules are silent on how far a flying that takes on a sudden weight would fall… I would probably treat it like feather fall, without the damage immunity. So 60 ft. per round. If the fall would deal damage to the flying creature, I would give it an immediate check to escape the grapple, just like maneuvers that put creatures into dangerous positions.
2) I would say that tripping only applies if the creature is touching the ground.
Where is the rule barring flying with a med/hvy load? I believe that is a 3.xism.
Load Penalty would apply penalties to fly skill (and acrobatics, etc)
but AFAIK there is no barring of flight, either magical or natural... But I could be wrong... Source?
If the rules are silent on distance fallen, it is the same as normal: 500' first round, 1200' every round after.
(that is from 3.5 FAQ, rounding from real-world acceleration numbers in normal air densities)
Grappled creatures cannot move, so the only option is Hovering.
That is kind of weird per RAW: Hover DC is 15, but in the 'Try Again' section of Fly (the only part that addresses failed checks in general), it says " If you are using wings and you fail a Fly check by 5 or more, you plummet to the ground, taking the appropriate falling damage (see Environment)." Which is clear enough if you fail by 5 or more, but what if you fail by less than 5?
I think that line makes most sense when applied to normal Fly checks which besides Hover are all achieving movement and the check is just to maneuver extra-fancy: if you fail by less than 5, you are just obligated to move in the normal non-check manner - if you can't fulfill that i.e. move normally, then you would then be trying to Hover. For Hover, if you fail the check at all, you should plummet to the ground (see above for normal fall speed/distance) unless you DO have the option to move normally and can do so when your attempt to Hover failed (unless you failed Hover by 5+ in which case you will plummet no matter your other normal movement options).
High Wind Speeds are just an additional limitation/repurcussion outside of flying itself, you will still need to make a Fly/Hover check, to which the Fail by 5+ rule applies, but it shouldn't DOUBLY apply to the High Wind Speed checks themself (IMHO, although there is no exception for them per RAW).
Also note about the 'Avoiding Falling damage' option: " You cannot make this check [to avoid falling damage] if you are falling due to a failed Fly check or a collision." So once you're plummeting to the ground because of failed save, you are going to fall the full normal distance, barring feather fall/etc.
Of course, 'grappled creatures cannot move' can also be read as preventing ALL movement, e.g. falling, a mount moving with you, bullrushes. In that case, you would float in mid-air as long as you're grappled regardless of all other considerations (i.e. neither grappler having any flying ability at all). I go with the narrower reading of the movement restriction though, which only prevents your own movement (allowing bullrush, gravity, etc).
Of course, there is still more problems with grapple if we go with the narrower restriction, since grapple isn't immunizing a non-flying grappler from falling just because the target can fly/hover, and vice-versa if the grappler can fly but the target ends up falling (because they fail hover check or can't fly in the first place), nothing prevents the target from falling while the grappler stays in place.
Tripping applies if you are not Flying... In this case, as soon as you are falling it would work.
Note that you can fall by 10' when attacked and fail a DC10 check (and falling provokes an AoO), so that is a window of opportunity for Trip AoO.
But once Tripped/Prone, you still can Fly, you just have the movement penalties until you right yourself/'stand up' per normal actions for that.
Supposing a monk is fighting a natural flying creature
1) If the monk succeed a grapple check woulds that make the flying monster (and the monk) to fall?
It wouldn't necessarily make a flier fall.
The grappled condition means it can't move. Just as a biped walking doesn't fall if grappled neither should a flying creature. That said, at the end of the flying creature's turn it must either escape the grapple or make the requisite Fly skill check of DC15.
2)IF the flying monster can not fly while grappled would that means the monster losses his immunities to being tripped?
If the flying creature fails the Fly skill check to hover then it must land. The now landed creature is no longer flying and would therefore be a legitimate target for a trip attempt.
Grapple, as part of the initial success, pulls the target adjacent to the grappler. He can use further grapples to move the grappled target and himself and place the target in any space adjacent to himself after moving that he wants. So there's always that.
I do think if grappling and only the target can fly, the pair would fall if weight limits are exceeded. Default would be both take falling damage; a houserule to have some opposed grapple check to force one or the other to act as a crash mat and absorb some/all the damage the winner would have suffered (on top of loser's own falling damage) would be fun.
Trip... in 3E, you could use it to cause flyers to stumble and stall. But PF just says they're blanket immune to trip. *sigh*
Where is the rule barring flying with a med/hvy load? I believe that is a 3.xism.
I located it recently. It's not clearly stated in one location, but…
Barding is a type of armor…Flying mounts can't fly in medium or heavy barding."
A medium or heavy load counts as medium or heavy armor for the purpose of abilities or skills that are restricted by armor.
Now, RAW, all that means is that mounts can't fly while wearing medium or heavy armor, or carrying a medium or heavy load. As none of the base races can fly by default, most, I think, would interpret that to all mean all creatures rather than just mounts.
As far as falling, the 3.5 FAQ doesn't qualify for a Pathfinder ruling, but whatever works in your game is fine. I just figure a feathered creature will have some sort of benefit perhaps similar to a spell called feather fall (but not the whole benefit, obviously).
EDIT: Ninja'd by MacGurcules! It would be nice if future releases included something about this under the Fly skill itself.
I asked James Jacobs about this back in March.
This should answer all your flying encumbered questions.
Edit: linked to my question - his reply is just a few down.
James Jacob's answer is a good one. Barding impedes natural flying since the medium/heavy armor keeps the wings from flapping. But if the same flying critter picked up its barding in it's claws/talons/whatever, then it could still flap and still fly, though maybe slower if the weight reduced its movement speed. That all makes total sense.
I suppose special medium/heavy barding could be made that leaves wings unimpeded, but normal barding has no such provisions. Creating a set of Full Plate Griffon barding that fits a griffon but leaves its wings free to flap could probably be done, with DM's permission, but the existing rules don't have a provision for it yet (that I know of).
For the OP's questions:
1) If the monk succeed a grapple check woulds that make the flying monster (and the monk) to fall?
No, nothing in the rules explicitly stipulates that you must fall when grappled. One might infer that if you cannot move then you cannot fly, but this can be argued by using the rules for hovering, in which case, not moving means hovering which requires a Fly skill check or the Hover ability - but even then the rules only say the creature plummets to the ground if it fails that hover check by 5 or more.
I think as a DM, I would rule that a grappled creature can hover. If it initiated the grapple, it can attempt to move if it successfully maintains the grapple, as per all the stuff Grapple says. But if it is pinned, now I would rule that it cannot fly (naturally, with wings) while pinned. This makes sense to me and it doesn't violate any RAW, but it also isn't RAW either.
2)IF the flying monster can not fly while grappled would that means the monster losses his immunities to being tripped?
Trip rule: "Some creatures—such as oozes, creatures without legs, and flying creatures—cannot be tripped."
Flying creatures are not automatically immune to trip. Let me clarify. To be a "flying creature" you have to be actually flying - just having a flight speed is not enough. A "flying creature" standing on the ground is not, at the moment, a "flying creature" because it is instead a "standing creature".
If the authors had meant to give automatic immunity to trip to everything that is capable of flight, they would have said "Some creatures—such as oozes, creatures without legs, and any creature with a flight speed—cannot be tripped."
But they didn't say that. So you can, in fact, trip a creature capable of flight, but only when it is standing (not flying) at the time you try to trip it.
Furthermore, though it isn't explicitly stated by RAW so this is more of a DM ruling than anything stipulated in the RAW, falling creaures have a similar immunity to being tripped, for the same reason - their legs are not being used for support. Besides, what could be gained by tripping a creature that failed a hover check by 5 or more? Either way, it's falling.