Why obsess over Pip? Because he's funny and entertaining without compromising Hellsing's dark and serious overtones. And that's a good thing.
He's more relaxed than just about everyone else in the organization, more able to show emotion. He freaks out when Seras throws him across the room with a flick of her finger; he gripes when he's housed in a cheap motel room on a mission while Alucard and Seras get first-class accomodations; he laughs and jokes and sings dirty songs. He's open about sex and fear and all the things the other characters tend to keep locked up.
Although he's the comic relief, Pip isn't just there to act silly and let off tension. He's the captain of Hellsing's forces, and he's good at it: in terms of strategy, he's always a step ahead. When Seras and Alucard are ambushed and cornered, he shows up with he helicopter. When the mansion's under ground assault, he's already planted the mines.
Pip has his neuroses, of course; but they're not as overwhelming as the other characters', and he seems the most comfortable with himself - in fact, he's probably the happiest person in Hellsing. This happiness is good for everyone, most notably Seras. He knows how to have fun, something that she was in danger of forgetting.
And under this air of hedonism is something deep and noble. Yes, the Wild Geese kill for money. But despite this impersonal job, Pip values people; he has a deeply rooted sense that good, innocent people should be protected, and he'll die for it. That's worthy of respect.