I would, certainly. Other DM's might not, but for my own style, absolutely. There's nothing in his behaviour listed above that violates the code as written. If I may...
SRD said:
A paladin must be of lawful good alignment and loses all special class abilities if she ever willingly commits an act of evil. Additionally, a paladin's code requires that she respect legitimate authority, act with honor (not lying, not cheating, not using poison, etc.), help those who need help (provided they do not use the help for evil or chaotic ends), and punish those that harm or threaten innocents.
So, under the assumption he's lawful good and his only sins are whoring, drinking, swearing, and cynicism. That's fine by me, and as written, it's fine by the code.
Realistically, even in the clean-scrubbed version of the middle ages most D&D games are set in, a woman of peasant background working in a brothel is probably doing better than she would be married to Tomas the miller. So, provided the brothel works less like a slave ring and more as a business where a woman can make a better wage, if she's willing, there's nothing evil about solicitation in and of itself. While chaste is typically applied as a requirement for paladins, it's not in the RAW; after all, the gods of good want humans to be happy. That's why they're good. I've always felt it was more on the context of "a serving paladin should not wed or produce heirs to distract himself from his duty", and that as a church restriction rather than an article of faith. So long as brothels are legal in the majority of civilized nations and not specifically opposed by the church, and he keeps paying (especially since the madam tries to insist he not pay).
Drinking. Nothing evil about it either, presuming his church doesn't believe that all alcohol is dangerous. As long as he's not getting falling-down, puking in the gutter, soiling himself drunk on a regular basis, no major transgression is occuring; and let me say I don't think "Had one too many ales" should ever cost a class their abilities, regardless of how many roleplaying restrictions they tried to balance mechanical advantages with.
Swearing? That's so minor it barely worth consideration, so long as he's not swearing oaths by the fiends of hell.
Now cynicism...repeat after me..."Lawful Good does not equal Lawful Stupid". He's very, very aware of how the life of a paladin ends; in screaming agony at the hands, claws, or other offensive appendages of some horrible spawn of evil, and he's not very happy about it. That seems fair; nobody wants to die screaming in agony. Your average paladin is played in such a way, IMXP, that he has no idea what's coming for him. He never considers doing something else with his life, he's committed to this path and firmly believes his faith will win the day. Now Cedric here, he's firmly committed to his duty as a paladin in spite of the inevitable outcome of this duty. He -knows-, inherently, that he's screwed, and his best chance is a decent afterlife. I'd say, if anything this is more noble. He's committed to fighting, and dying, for the cause...for no better reason than it
SHOULD be fought for.
So, simple question - yes, I'd allow him in my campaign. Complex question - without house rules dictating greater restrictions on the paladin, there's nothing in the paladin class as written that forbids this particular concept.