Yes, but it would take your roc a move action to pick you up. There is no way within action economy for the rock to move, pick you up, and continue moving. It will have to end it's turn adjacent to you/pick you up and move next round.
They would be unable to carry anything else in their claws or attack with their claws.
If you were too heavy they couldn't carry you.
Yes, the Roc has the fly by attack feat which means it can take a standard action (pick you up) during a move, but if it does so, it cannot take a second move action.
So with a fly speed of 80, the rock could move say 30, pick you up and continue moving up to another 50, total move is 80. It could not then take a second move as it has now used both a move action and a standard action.
Once you are being actively carried, you and your companion would lose defences as if you were grappling.
The reason I as a GM allow the pick up as the standard action is it would in my mind be a kind of touch attack to grab hold - some GM's might even go so far as to make an attack roll, but for cinematic reasons I would simply allow it as you would not be resisting.
Even if picking up is a move equivalent action, recall that a standard action can be replaced by a move action - that's how you get to move twice a round. So, as you are spending a standard action in the flyby, I don't think there is any reason in the rules to stop this with the exception of encumbrance and the age of your companion...
Also, note that a starting companion Roc has a STR of just 12 when it starts out (on the bottom of the Roc page linked above) - so your GM might call for an encumbrance check which might affect if it can actually continue to fly with how much you and your equipment weigh.
As the Roc companion gets stronger, this will become a non-issue but at very early game levels, it could be a real problem.
Hope that helps.
It's also a free action to be handed something. Though it sounds very silly to say this, could you in essence hand yourself to the Roc - Be in position, put your weight into it, etc?
Surely the great thing about a Roc companion is that at the start (or just before) of a battle youve already hopped on its back!?
Yes, the Roc has the fly by attack feat which means it can take a standard action (pick you up) during a move, but if it does so, it cannot take a second move action.
The animal companion Roc does not automatically have the fly-by attack feat. Though, it could be selected.
Please note that the animal companion version of creatures is not the same as the bestiary entry of the same animal.
Rocks have feathers?
But yes, it does seem like giving your animal companion Fly-By Attack (which I initially thought only applied to attacks like Spring Attack, though it turns out it doesn't!) is the way to go here.
Alternatively, you could possibly ready an action to mount your roc as it passes by you, but I imagine there may be a Ride check involved to climb onto a moving animal.