1) I'm not sure that the general public's tendency to evaluate the "character" of political figures is always a useful heuristic on which to assess either shared values or especially eventual policy positions. In fact, I'm pretty sure it's practically useless. That said, there is some validity to the notion that someone who is at least slimy and suspicious in their personal lives has some slime and suspicion to their political views. It is just not nearly as useful a notion as is commonly applied.
2) If Gingrich did ask his then wife for an open relationship, a) that's not a new news story (it appeared in an Esquire story over 2 years ago) and b) it's not particularly troubling morally and ethically by itself as some sort of salacious detail.. The troubling part is that he asked AFTER he had been carrying on an affair for several years, and after they were already married. More people approaching this question in the manner of Newt or other routine philandering types should look to either a) not get married in the first place or b) be open about the possibility and likelihood of their philandering ways with potential spouses who could then make something approaching a rational choice to either accept, moderate, control, or reject this behavior.
3) The actual significance of it for Newt is that he is marginally infamous for his public attempts to castigate other political figures for their "moral failings" (ie, affairs and infidelity). There are a number of GOP figures (Mark Sanford, David Vitter, etc) for whom this is a strange position as a result.
4) To me, there are numerous ethical failures relating to an over inflated conception of the offices of government as laid out in the Constitution and a respect for said institutions that are far more troubling than how he ran his private life. For example his pandering on "activist judges" and subsequent calls to reign in the independence of the judicial branch, one of the hallmarks of the American system of governance (which is to say nothing of the meaninglessness of the term "activist judges" in the first place). Or his idiotic conception of freedom of religion that somehow does not or should not apply to Muslims, or should do so only within some constrained version of international reciprocity (which is a perverse way to apply international laws in order to restrict human liberty). And so on (embrace of torture, use of civilian review boards for immigration status with an implied emphasis on their religious affiliation being a litmus test).
This is to say nothing of his egomaniacal attitude toward... other human beings generally, or his ethical violations while in office in Congress, or his gross defense of his lobbying, and subsequent "pious baloney" that he somehow represents the Washingtonian outsider in this race.
The one positive benefit that could be derived from a Gingrich nomination is that he is likelier to be defeated than Romney (presuming economic conditions remain roughly what they are, on a slow growth trendline) and that the politics of ressentiment will have to subside rather than dig in. While this would be beneficial to libertarians and progressives alike, for reactionary conservative politics to receive some form of comeuppance and rejection by being utterly repudiated in the polls, I'm not sure that we want someone like this, this close to the Presidency to receive a major party nomination. Nixon might have been the last such individual, and even Nixon had the sanity to have his (numerous) insane moments in the privacy of his office, as recorded, rather than airing them flagrantly and publicly reveling in his own stew. Also, it didn't work out so well for the country to give Tricky Dick the reigns.
There's some fortune in the amount of disdain the general public has for Newt, but I'm not sure that a) the GOP is that stupid to put him forward as the candidate. The base might be, but I doubt they have no sharp operators at all over there to work to prevent it. Romney's negative ads worked, and the dogpile of Paul or Santorum against him when he has been a front runner is likely to add to the flames that are so easily started around his campaign. Expect if he wins Florida for the fires to start in earnest to prevent his ascension b) his debate performances will likely not work in a general election. The GOP line is that Obama is stupid and would be easily defeated by an "intellectual" like Newt. That is a gross miscalculation of one's opponent. It's also only possible because of Newt's overinflated sense of intelligence and reputation (ill-gotten) as a man of ideas. I'm not sure that any of the GOP field has possessed the intellectual subtleties to win interparty debates consistently, but Newt is hardly distinguished among said field. He hasn't been proposing new and innovative ideas. He's just been playing the "media is evil liberals" card over and over again. The general public doesn't eat that up the way the Faux audience does. It also doesn't want moonbases or overturning child labor laws (no matter what well-intentioned basis is used for doing so, which Newt does not have one). These are not transformative and powerful ideas. They're the stuff of Bond villains or bad Indiana Jones movies. The biggest flaw in the GOP if it were to put this man forward is in the old maxim: "It is said that if you know your enemies and know yourself, you will
not be imperiled in a hundred battles; if you do not know your enemies
but do know yourself, you will win one and lose one; if you do not know
your enemies nor yourself, you will be imperiled in every single battle." They would not know themselves, nor understand their opponents. Victories in such contests are possible only through extraordinary circumstances (ie, "luck").