Every kind of Pokemon. The safari zone was designed to bring trainers the thrill of every region, so it contains all Pokemon, except for the new Unova region Pokemon.
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to the next gym leader(Thats kind of obvious)
You can capture it in the safari zone but it's kind of rare.I don't have all info but some pokémon sites probably says how to make it happen.
You can try to get the other kind of orb which will get you kyogre and after you get kyogre you can go back there and get rayquaza.
It depends on what kind of egg it is. It takes a certain time for each Pokemon. Togepi eggs are particularly quick to hatch.
A flying-type Pokemon.