Note the order of the attacks, the bite comes before the claws. If the roll for the first claw hits and that's applied to the bite, then it also hits, even with cover because it's a primary attack and thus at +5 more. And yes, he'll grapple as soon as he hits with a bite or claw.
OOC (not spoilered since it won't affect the fight): As for buffs, he just has a large number of overlapping prot goods because there was nothing else. Another case of stupid spell selection. Every single dragon with spell selection should ALWAYS get the following three 1st-level spells: shield, mage armor, shield of faith. No ifs, ands, or buts about it. Likewise, they should (almost) ALWAYS get bull's strength, cat's grace, and bear's endurance at second level. The spell selection for Drac is terrible. Ventriloquism? Darkness? Obscure Object? Ridiculous! Prot. from Good and alarm are nice choices. Magic missile or any other SR-based spell sucks for dragons until they get really high or get assay resistance.