For the first question, I'm trying to find out if there was something that was done in lieu of antibiotics (since they hadn't been invented yet) that possibly raised the odds of surviving amputation? Perhaps cauterization that killed most of the bacteria or something of the sort?

For the second question, how did amputees get home after their amputations? Many battles took place in remote locations, so how did soldiers who had amputations (especially leg amputations) get back home while the rest of the army had to march on? Did they have people who were tasked with getting the injured soldiers to the nearest train station? Did people drive them home in carriages? Or were they just given a set of crutches, some well wishes, and set on their way to try to make it home by themselves?

More of course can be said here, but i wrote previously about medical care in an old AMA which might be of interest.

Dr. Samuel Gross wrote a book called Manual of Military Surgery in 1861. Chapter VII is titled "Ill Consequences of Wounds and Operations" which will hopefully be helpful.

To treat a traumatic fever Dr. Gross states

The treatment must be exceedingly mild; the patient will not bear depletion, but will, notwithstanding his fever, probably require stimulants and tonics, with nutritious food and drink [...] A gentle anodyne and diaphoretic mixture, as morphia and antimony in camphor-water, may be needful, in the early stage, to quell the fictitious excitement or attempt at overaction.

For muscle spasms as a result of the surgery he prescribes

anodynes in full doses, with a little antimony, the use of a moderately-tight bandage, and warm-water dressing, medicated with laudanum and acetate of lead

For "Profuse suppuration" (pus formation) he states

quinine, iron, cod-liver oil, and brandy, with frequent change of dressing, cleanliness, and ventilation.

Onto various infections. First, he discusses Erysipelas (acute skin infection)

Should be treated with dilute tincture of iodine, or anodyne and saturnine lotions, quinine, and tincture of iron, with nutritious food and drinks.

Gangrene

The proper remedies are sequestration of the patients, the free use of the nitric acid lotion, iodine to the inflamed skin, charcoal, port wine, or yeast cataplasms, and frequent ablutions with disinfection fluids, aided by opium, quinine, tincture of iron, lemon-juice, and other supporting means. Mopping the affected surface freely with strong nitric acid often answers an excellent purpose. The favorite remedy of Pouteau was the actual cautery.

The third type of infection he discusses is Pyemia (abscesses due to sepsis basically)

It was the great source of the mortality after amputations, especially secondary, during the war in the Crimea. [...] The treatment is essentially the same as in erysipelas.

Finally, he talks of traumatic tetanus.

Traumatic tetanus is not very common in military practice. [...] The mortality from traumatic tetanus is notorious. Hardly one recover. Nearly all perish in two or three days from the attack. The most reliable remedies are opium, in the form of morphia or acetated tincture, in large doses, in union with camphor and antimony. The effects of Indian hemp are uncertain. Chloroform with mitigates pain and spasm. Amputation, except, perhaps with the wound affects a finger or toe, will be worse than useless, as will also be counter-irritation along the spine. To prevent the disease should be our business, and to do this no wounded person should ever be exposed tot he cold night air, or to currents of air at any time. After all amputations, however trifling, special directions should be given upon this point.

(emphasis is Dr. Gross's).

So what does all of this tell us? That, as /u/heyrainyday stated, infections do not guarantee death. Combined with available medications and treatments, people could often survive these encounters with infections. That said, there appears to have been concerns over giving medications out too freely as he states

The use of heroic medicines, or of any medicines in large doses, in these diseases, and also in cases of severe wounds, cannot be too severely reprobated. More men, there is reason to believe, have been killed in this manner in the armies and navies of the world than by the sword and the cannon. Let medicines, then, be administered sparingly.

That said, by the time of the Civil War there was an understanding, at least in basic terms, of the relationship of cleanliness and infections. Dr. Gross states

Considering, then, the frequency of the occurrence of these diseases, and their excessive fatality, it behooves the military surgeon to use every means in his power to guard, in the first place, against their outbreak by the employment of proper hygienic or sanitary measures, and, in the next, to treat them with all possible diligence and judgement when their development is unavoidable.

In fact, on the next page, he recommends frequently bathing patients to keep them clean.

Of course, much of what Dr. Gross is writing about is ideals. How to ideally deal with infections and diseases, and war is far from ideal. But, that's what they were looking to do in order to keep infections/disease at bay either initially, or after the patient is sick with one.

Here is a PDF version of Dr. Gross's book, if you wish to peruse it. I also recomend "Bleeding Blue and Grey" by Ira Rutkow.