The Cases for Other Pairings
Even though Hellsing isn't a romantic series by nature, fans will be fans, and that means fans will ship assorted pairings. Alucard and Integra are both paired with just about everyone in the series by some fan or another, and all the characters are mixed and matched in plenty of different ways.
On this page I'll give a brief hearing to the more popular pairings, and explain my take on each.
Seras/Pip - The One Canon Pairing of Hellsing
This actually happens. In the manga. There's no way to discuss how we could be reading too much into things and seeing romance that isn't there. It's there. It's blindingly canon.
Besides, it's an awesome pairing. Pip the dashing rogue gets timid Seras to stand up for herself (by smacking him when he annoys her); sweet and pure Seras brings pervy Pip to experience a real, honest love beyond lust. It works out wonderfully.
So I'm not even going to entertain other pairings of these two characters on this page. Some have their value, and some are possible in the absence of this one, but SerasxPip has taken place and there's nothing to be done about it ^_^
Alucard/Walter - Could have been
These two are old friends, in the most literal sense of the word. They've fought alongside each other in a war; they converse easily and freely; they trust each other. And although Walter's usually perfectly satisfied to play the quiet butler, he's mild-mannered after the fashion of Clark Kent: you really don't want to have him mad at you. I can see Walter and Alucard having had a relationship around the time of Hellsing: The Dawn (taking place in 1942).
Besides, Alucard looked seriously like a girl then.
But that was then and this is now; today Walter prefers the quiet lifestyle. Whatever they were in the past, Walter and Alucard today are just friends - old friends, close friends, deep friends, "best friends" even - but not in love.
Integra/Walter - No way
These two, quite obviously, love each other. They would fight and die for each other; they're perfectly at ease with each other; they trust each other completely. Walter's quietly amusing comments got Integra to smile in the final episodes when everything was falling apart. Integra has only ever suggested giving up when the alternative involved putting Walter in danger and possibly losing him too. Walter is devoted to serving Integra.
But it isn't a romantic kind of love. Not at all. It has some elements of a father-daughter relationship, in that Walter helped raise Integra and certainly loves her as much as he would his own daughter. But then, Walter unhesitatingly accepts Integra as his boss and his superior, the one he follows unquestioningly. It's a unique and wonderful relationship, but it isn't a romance and never will be.
Alucard/Anderson - Er, perhaps
Well, some qualities that Alucard admires in Integra (strength, stubbornness, and a refusal to give up in the face of overwhelming odds) are certainly present in Alex. But that's really where the similarity ends. Alucard holds Alex's self-delusion to be silly; he enjoys playing with the guy, though, in a cat-and-mouse type of way. I suppose you could work some sexual tension into that.
As for Alex, he hates Alucard with an unnatural fanaticism, even for Section XIII members. But even when he overcomes his urges to kill Alucard, it isn't as though he suddenly starts liking the guy.
There's a lot of strong emotion (hatred) flowing between this pair, and one could add sexual overtones to their struggle for dominance, but that's not neccessary, and the people who do this are usually yaoi fangirls whose time would be better spent with other, more substantial pairings.
Integra/Enrico - He so has a crush on her. Mmhmm.
There's a wonderful shrine at Eleventh Hour dedicated to the glory that is IntegraxMaxwell, and I really couldn't say anything that isn't there already. So I'll just link you to them and leave it at that.
Seras/Walter - Cute but not happening
I know I said I wouldn't do pairings with Seras or Pip, but I just wanted to point you in the direction of another pairing shrine, dedicated to the not-happening cuteness that is SxW:
Anybody in Iscariot/Anybody Else in Iscariot - Maybe.
Enrico, Alex, Yumiko, Yumie, Heinkel, and Father Renaldo can be mixed and matched in approximately 6,120 different combinations of pairings, not all of them equally likely. (YumikoxYumie - how's that for a split personality? Given that she/they are a split personality . . .)
We only get a few possible-pairing character interactions with these six, though. Yumiko is quite distressed when Enrico is injured in the third Cross Fire. Heinkel is able to seamlessly rein in Yumie's berserker rages; the two work together flawlessly. It's a stretch to extrapolate a pairing from either, but I happen to like those two.
Integral/Anderson - Interesting dynamic, but no
CAUTION: This analysis features spoilers like whoa. Only highlight if you're willing to have some big stuff (volumes 6 and on) spoiled for you!
In the sixth volumes of the manga, Alex, Heinkel, and Yumie have been sent - along with several thousand Vatican knights - to attend the conflagration in London and hit Hellsing when it's also trying to deal with Millennium's all-out attack.
Integra's ended up on her own, and this group corners her; she basically snaps them all into line behind her. Over the protests of Yumie and Heinkel, Alex tells them all that yes, they are going to walk Integra home.
They're supposed to be capturing her, mind you. This was Enrico's order, and he's now not only the boss of Iscariot, but an archbishop. He wields amazing authority. Alex kills him - stabs him through with bayonets - and has begun taking orders from Integra.
I'm still not sold on this getting romantic, much less more romantic than AxI, but it's a very interesting dynamic.
The Case Against
The Witness Stand