Hayate the Combat Butler/Headscratchers

How exactly are Sakuya and Nagi related?

It's said that they're cousins, but think about it: Sakuya, Nagi and Yukariko all have green eyes, which makes it seem like one of Sakuya's parents would have been Yukkyun's sibling, right? But Yukkyun was an only child. Oversight on the author's part, or evidence of a human cloning cover up conspiracy?

Just because they have the same eye colour doesn't mean that they have to be that closely related. It's possible for cousins to be related though grandparents, uncles, aunts, etc. Last I checked "cousin" was not solely defined as "child of a parent's sibling".
Maria states at one point that Nagi's grandfather is "your only ... blood relative." However, on the same page, Sakuya says he's her grandfather, too. How can Saku be Mikado's granddaughter without his other granddaughter having any blood relatives? One of her parents must've been his adopted child -- possibly disinherited for the fooling around that produced Gilbert, since nobody, including Sakuya, is surprised when Mikado mentions that when he dies Nagi will inherit everything.
Might be Related to Nagi's dad?
- Just because someone calls an old man "ojiichan" doesn't mean he's her grandfather, just someone in a grandfatherly position. So he could be her grand-uncle or something like that. Come to think of it, Aika called him "grandpa" as well.
- If Sakuya was blood-related to Nagi's father, she'd be a blood relative of Nagi as well, which still contradicts Maria's comment. The point about "ojiichan" not necessarily referring to a literal relationship is well taken, though.

Why is Segawa's brother(Kotetsu) her butler?

It's explained at one point that it's an old tradition for the elder sibling of rich families to be trained as a butler. Not too sure if this is just in-universe or not.

Will Hayate's parents ever get their due diligence?

I've only read up to Chapter 53 of the manga so far, but seriously, WHY HAVE THOSE TWO NOT RECEIVED THEIR PUNISHMENT FOR THEIR EMOTIONAL, FINANCIAL, MENTAL AND NEAR-PHYSICAL ABUSE OF THEIR OWN SON?! Never mind the fact that their actions are Played for Laughs, it's still cruelty to a child!

I won't say anything, but It Gets Worse
The above troper is right. Just wait until the The End of the World arc starting Chapter 178. You haven't seen anything yet.
Update from the original poster: To the two above posters, you needn't worry about spoiling me, as I already know about what's going to happen up to when Athena joins the main storyline, and I know about how her involvement kicked off the primary back-story (in fact, it was my spoiling myself that prompted me to actually read the manga to see just how it would all unfold)...but still, I'm upset that Hayate's parents apparently won't get their due diligence.
- At least, not yet.

Why is Athena considered a Base Breaker?

I finally caught up to the manga, but I don't see anything wrong with her. She's not a serious rival for Hayate's affections once that arc is over, so I don't get the hate. If there is any hate and it's not just some random troper's fevered imagination.

Pretty sure it's your imagination. She got second in the first popularity contest she was eligible for, after all. Sure, she lost by about a thousand votes, but Hina always wins by a landslide.
Actually she got 5th place in the first one she was eligible for, and 2nd place after that. She lost by about 2000 votes, which was still the first time anyone got over half of the votes Hina did.
Athena brought with her a seriousness to the story that wasn't there before. When she left, it dissappeared, with much of the fandom saying that HnG was no longer interesting.
Should I say Ship-to-Ship Combat?
It's hard to see how it happened unless you were there when it started. Back before the series was as popular and got an anime, the End of the World Arc wound up getting scanlated during the Shimoda Hot Springs Arc (about 80 chapters early, leaving a big annoying gap that wouldn't be filled for years). This was only made worse during the Greece Arc as we had to wait a week or more for each chapter, and believe me it was way more frustrating than you'd guess reading through the whole arc now. Some people only started reading because of the seriousness (and those people therefore tended to like Athena) while others didn't like seeing our wacky lovable cast go through the emotional turmoil. It definitely didn't help that Hinagiku, hands down the most popular character, got shafted quite a bit and led to many, many arguments on the forums between Hinagiku fans and Athena fans. The shipping wars made things worse, but Athena has been a pretty controversial character ever since we found out who she was.

What does Isumi meant by saying "Well, I think it would be a problem to Nagi if someone much closer to Hayate-sama than I am were to fall for him."?

I've been re-reading the manga again, and in volume 04, chapter 34, where maria and Isumi were talking about Hayate and also Nagi and Isumi's friendship, and in the end Isumi stated a line that caught my interest: "Well, I think it would be a problem to Nagi if someone much closer to Hayate-sama than I am were to fall for him." (Page 42, I believe)

I was wondering many times about who is the person she's referring to. This question really bugs me.

I thought she was talking about Maria?
Another prediction of Hayate falling for Hinagiku? Thus closer to Hayate's affections than Nagi, or even Athena, since she's the only one that Hayate has actually confessed affection for.
- Wait what how and when?

Does Hayate really have an excuse to be poor?

Rule of Funny nonwithstanding, it seems obvious that Hayate could remove his debt in a millisecond by hiring himself as some kind of international contract agent/enforcer of sorts, courtesy of his superpowers. His first plan upon "going bad" is to... hold someone for ransom. Well how about working as the world's deadliest assassin or turning back and taking over the Yakuza he just ran away from--or if that goes too far down the Moral Event Horizon, how about stealing priceless treasures from vaults around the world not prepared to deal with superhuman assault. Seriously--you can do a lot with superhuman strength, agility and resilience, and the fact that Hayate remains in Perpetual Poverty is just testament to how Too Dumb to Live he really is.
Dude, Rule of Funny.
- Plus, the fact that he's a Dogged Nice Guy would really cut into any chance of him being an effective assassin/thief/Yakuza boss/criminal-what-have-you.
- He also seems to only really activate that mode when his master (or someone she cares about) is in danger, making him less than effective as anything but a bodyguard.
He could have removed his debt in a millisecond by filing the appropriate papers and declaring bankruptcy. This wouldn't fly with the yakuza, but it would have gotten him out of Sanzenin servitude. However now that he's essentially cost Nagi her fortune that would be a bit of a dick move, so I don't expect him to ever do so.
- Well, considering his luck... And for a more serious answer, it's possible that, for him, it "feels right" to be a butler, and might see it as some sort of chance to make up for his supposed Failure Knight status to Athena. Additionally, it was a chance to repay Maria for "saving" him. And really, it's not like he disliked serving Nagi. He could've filed the papers but maybe he just didn't feel like he had to, or wanted to.
Okay, this series totally deserves a sequel called "Hayate the Bounty Hunter."
Actually, yes, he does have an excuse. On the very beginning he was trying to work while also going to school, meaning that he could only do low-pay jobs, and that was after lying about his own age. He could've done better-paid jobs for Yakuza, but that would require him to shed his naivete (which, as we all know from the "End of the World" arc, stems from the trauma applied by his parents throughout his childhood, and later also from being "abandoned" by Athena) and "poor man's mentality" (which is the reason why so many people who were poor in their past are incapable of earning enough money to stop being poor; they are mentally incapable of managing money and allow it to easily "go away" [a trait evident in Hayate], and they are content with the level of wealth they have if they have anything [a trait also evident in Hayate] which reduces their motivation to earn money to nil). So yes, to sum it up, Hayate's excuse for being poor is the mental trauma his parents caused with their irresponsible behaviour.

Just how many people knew about Hina's love for Hayate?

I'm simply curious. Surely it's not just Ayumu.
It's seems like it's pretty much obvious to much of the main cast, actually.
I believe Maria, and possibly even Isumi, knew it would happen before Hinagiku was even introduced as a character. From both sides actually.

Anybody want to make a guess of how Hayate's reaction will be if Hinagiku confessed to him?

Because, the issue between them was the one that is the most intriguing to me.
- A Asteroid would probably crash down on him or something.
- Hayate: "Ehhhhh?!" Then Status Quo Is God.

Dangit, why does my heart insist on shipping Ayumu/Hayate when my brain knows that the only way they're liable to actually get together would be through the Ayumu/Hayate/Hinagiku Tenchi Solution, which won't happen unless Hinagiku finally mans (girls?) up and tells him her feelings? Gaaahhh... poor Nishizawa.

But seriously, you gotta feel sorry for her since, with the way the story has been unfolding, she's practically all but doomed to end up an Unlucky Childhood Friend... I dunno, but I can't help but root for her since she's the most normal one out of Hayate's harem.
- You have my sympathies. I'm in the same situation with my Athena/Hayate pairing, even though I know there's no hope now the Greece arc is over and the author seems hellbent on shelving A-tan until everyone forgets about her. Feels bad, man.
- That's because you simply like the girl for some reason. Personally I'm more of a Hayate/Athena person (though I wouldn't mind Hayate/Hinagiku, or even any other pairing), but I am aware that Hayate won't end with anyone. He's just too nice to choose, and it'll become even more difficult for him when the size of his wannabe-harem increases. So yes, Status Quo Is God, but this time it's kind of justified (I once experienced a mere Love Triangle, and it quickly went to Albuquerque with nobody being together because of all the awkwardness).