Loads of questions here so let's answer them then explain some things.

How many suicides were by firearm in late 1700s America?

Nobody knows as there weren't statistics kept of that, but it did happen.

How would a violent suicide have been perceived?

Essentially like a non-violent suicide (which was worse than we view them now). Blood and gore was not foreign to most farmers in South Carolina (where the movie is set) around the time of the Revolution. Family members and especially women (and those in high society) would be most taken aback by the violence of a violent suicide.

How common was suicide in general?

Not very but there are records of it. It has happened in just about every culture ever studied since the dawn of civilization. There is no indication it happened more often than any other time in American history if that's what you mean.

Would they have reacted in this way, just kinda going home for a week?

Not really what happened: Let's come back to this.

Was suicide still seen through a Christian lens, that it was sinful?

Oh yeah, and big time. There was no "mental illness" then, so it was viewed as either possession of evil spirits/satan or (and literally in the law books of S Carolina as) the criminal act of self-murder.

Would the survivors left behind be ostracized or shunned in anyway because of this?

Yes, but not as much as the individual. There is a record of one in Virginian being buried at a "cross path" so his grave would be forever trampled. In the Southern Colonies (NH and Southward) if was law that your possessions would be seized and sold at auction and you would not, by law, be allowed a Christian burial. This started in Virginia and spread from there, heavily influenced by similar English laws predating the colonies by 1000 years. Massachusetts, the source of most legal influence further north, wasn't too different - they outlawed it in the mid 1600s.

What was the most common way to commit suicide if not by firearm?

Hanging would be my guess, though again we have no statistics of this.

About that week off question: rewatch the scene. It wasn't meant to give them time to recover from the death of a friend but rather to secure their own families. John Billings shot himself after finding his wife and child dead from the British attacking civilians in villages as retaliation for the militiamen's acts. The men were dispersed to go and protect their homes from similar attacks/move them to safety (iirc Martin's kids are sent to Aunt Charlotte around that point). This was an attempt to show the evil of Colonel William Tavington who was very loosely based on a real man, Colonel Banastre Tarleton. Folks in the south, particularly S Carolina, saw Tarleton much how Georgians saw Sherman 100 years later - as a butcher and war criminal. He was heavily vilified for several things but this portrayal enacted major backlash from British media, alledging he had been improperly represented - and that is true (I mean burning a church full of people? Really? Come on now, guys, let's keep some realism). So he wasn't really the bad guy we remember him as (though he wasn't particularly nice to us, either). He also didn't die in battle but instead reached rank of General and outlived the character Mel Gibson's Benjamin Martin was primarily based on, Francis Marion (though elements of several others were mixed in). But none of that makes emotional drama so elements were combined or created entirely to make a blockbuster movie with massive factual inaccuracies. They needed a black knight and a culminating scene wanting for revenge (which was a feeling towards Tarleton by S Carolinians) so the white knight may save the day and exhibit to all he is The Patriot.