I have an earlier answer to your second question!

Frank (Frances) Folsom's age at her marriage to Grover Cleveland was something noted by the press, absolutely. However, any criticism seems to have been rather mild.

One article, printed in multiple newspapers including (here) the Indianapolis Sentinel, merely said:

She is the daughter of tbe former law partner of Mr. Cleveland, and is spoken of as a lady of great brilliancy, and one who would be an ornament to the White House and to society as well.

She is perhaps a trifle young for an old fellow of forty-eight, for as the President celebrated his birthday on Wednesday, it is hardly worth while to try and conceal his age longer.

Her age, if so delicate a subject may be touched upon, is probably about half that of the President.

She is spoken of as very handsome...

Even when a newspaper column mentions her age, there is nothing said about their earlier relationship. I'm especially a fan of this clip from an Iowa paper I found online, in which the author discusses Frank's childhood handwriting and even her childhood nickname with nothing but approval for the woman...and comments (apocryphal or not) about how her handwriting was setting the trend for local women.

Grover's 1923 biographer, Robert McElroy, repeats an unsourced, possibly apocryphal story about the president's reaction to the rumors circulating in advance about his possible marriage...to Frank's mother:

I don't see why the papers keep marrying me to old ladies all the while--I wonder why they don't say I am engaged to her daughter.

I am not sure I would take this particular story seriously, especially because McElroy doesn't cite it beyond "an old friend and her daughter." On the other hand, I've never seen any disagreement that the press did expect Grover to marry the widow Emma Folsom, not her daughter.

This makes some modern scholars' argument that Grover began a romantic relationship with Frank and (especially) married her as an election/reelection strategy...hm, interesting. On the other hand, that such an argument can be made in the first place likewise points to a general public acceptance.

So you can see the marriage did not quite seem the most natural, but wasn't a widespread problem. The media's utter infatuation with Frank herself might have helped matters, as probably did the fact that Grover had already beaten one sexual scandal (a) by owning up to it, and (b) it still being less bad than the competition's political sins (c) probably because the parts of the sex scandal that reflected truly poorly on him depended on the word of a woman, and people in Cleveland's era were very concerned about accusations by women ruining men's lives.

And by the time Frank Cleveland was departing the White House along with her husband, "a trifle young" had become a good thing:

A modest, robust, enthusiastic girl she entered the White House. The entire nation was interested in her. It delighted in her youth and beauty.

...It is, however, noteworthy that eleven years later, the Chicago Chronicle was remarking not just on her youth at the time of marriage but remembering that people had talked about it.

To your first question, he probably did. To your second, not really.

The best scholarship on this comes from a very odd source. Charles Lachman, the executive producer for the voraciously tabloid Inside Edition television show and one-time reporter for the New York Post - another news source not exactly known for highbrow research - actually wrote a book on the subject, A Secret Life. While Lachman has no formal academic training in history and his writing definitely shows it when he strays into larger conclusions about the era, he's actually not done a bad job in his research, discovering and using substantial primary sources that other historians have largely ignored. While the area is in definite need of additional work by trained academics, the result is a pretty damning tale.

Cleveland - related to General Moses Cleaveland, the founder of that city (the name changed as a newspaper editor thought Cleveland looked better on the masthead) - ended up in Buffalo largely by accident: on the way to his namesake's city, he stayed with his uncle who convinced him that going further west with neither contacts nor career plan was a disastrous idea and that he should end his journey there. Buffalo in the 1850s was a wild place; as the terminus of the Erie Canal and major distribution center for places West, a large and wealthy merchant class developed - along with a world class redlight district for all the transportation laborers in town for a few days. (Indeed, you can make a pretty reasonable contemporary comparison to Gold Rush-era San Francisco.)

Cleveland used his uncle's contacts to get a position to read the law, became a reasonably successful commercial lawyer, dodged serving in the Civil War by hiring a substitute (who didn't fare well in the war and to whom Cleveland was fairly terrible later in life), and went back and forth between his commercial law practice and serving in the District Attorney's office, where the contacts there were quite helpful as as he likely was familiar with the red light district as not just a prosecutor but as a customer while he remained a notorious bachelor. (Lachman outright states he preferred the company of men, but also regretfully admits there's no real evidence that he was bisexual.) Cleveland then served as Sheriff and by most accounts did a fairly reasonable job, even if Buffalo had very loose enforcement of many of its morality codes and bar closing times along with Cleveland becoming wealthy from fees associated with the office. Oddly enough, this was not a sign of corruption; Lachman doesn't mention this, but office holders collecting large fees as part of their office was very commonplace for appointed and elected officials in 19th century America and is actually one of the major reasons why the politics were often so nasty since the money was just that good.

So the summary so far is that prior to his meteoric rise as a clean government reform mayor and governor in the 1870s and 1880s, Cleveland was a fairly wealthy bachelor attorney with significant and deep contacts in the law enforcement community. He also had a law partner whose child he doted on - she called him Uncle Cleve - especially once the law partner died in an accident when she was 11; he'd returned to private practice a year before following his term as Sheriff. Cleveland also mentioned to his sister somewhere around that time when she nagged him to get married that first he "didn't think he'd do it" and then a few years later that he was "only waiting for his wife to grow up." There's no evidence whatsoever that Cleveland took a sexual interest in Frances at that time and it may very well have been an outright joke to an annoying sister, but given the eventual outcome it is also one of the creepiest utterances of any President.

So into all this in December 1873 comes a very pretty widowed men's store clerk who lived a block or two away from Cleveland, who Cleveland had been courting for months, and who ran into him on the street and apparently asked her to dinner. Six weeks later she became pregnant.

What happened that night? Well, according to Cleveland, nothing, along with having operatives spread that she fit better into the redlight district than as a proper lady. According to Maria Halpin in an affidavit signed two weeks before the 1884 election, he raped her.

Rape convictions were almost impossible to obtain against wealthy white males in the 19th century, and in Cleveland's case it would have been even more impossible given his vast connections in the law enforcement community. However, while the election affadavit was largely dismissed by most historians as an election shenanigan - it's omitted in the 1923 biography that /u/sunagainstgold mentions, and it's probably fair to say that Lachman relies on it a bit more than he should - there's plenty of circumstantial evidence that Cleveland took an undue interest in the pregnancy and the child. This included getting the best obstetrician in the county to attend the birth and then raise the child for a year, along with later legal manuevers and contacts to silence Halpin, including getting her to surrender all rights to the child and put him in an orphanage, manufacture enough evidence to have her temporarily committed to a a lunatic asylum, and then ensure the child was adopted by the now childless obstetrician's family, along with settling with Halpin out of court for a ridiculously small sum.

The 1884 election was not a good time for this and other evidence to come to national attention without the color of politics - it was one of the nastiest on record - but it does look like there was some fire under the smoke, and while the evidence isn't conclusive, it's troubling and I'd tend to believe the allegation.

As far as Frances Folsom, given Cleveland's reputation as a lifelong bachelor, as /u/sunagainstgold notes, the general reaction seemed to be relief that he was finally getting married regardless of the circumstances. The other aspect is that the budding romance was starting to be known prior to this; when at college, Frances was noted by friends as appreciating Cleveland's attention and being genuinely excited about going to the inaugural - even if around that time Frances' mother finally realized that he was far more interested in his daughter than her and was somewhat less than happy about it.

But while there are a lot of things to say against Grover Cleveland, at least his bride does appear to have reciprocated his interest by the time he began courting her - even if how he got to that point is more than a little disturbing.