To answer a few of your questions...
My understanding is that aperture/peep sights improve upon open sights in three ways.
1) Improved vision/focus - I don't know the science behind it, but looking through a small hole (aperture) improves our visual focus. As an example, a person who normally needs glasses can punch a little hole in a piece of paper, look through it, and see as well as he could with his glasses. Same idea applies to shooting with aperture sights.
2) Natural sight alignment - Our eyes tend to center a front bead within a circular aperture more automatically than they do with with typical leaf or semi-buckhorn sights.
3) Longer sight radius - Most aperture sights are positioned at the back of the receiver or on the tang of the stock, thereby increasing the distance between the front and rear sights (compared to barrel mounted sights). The farther the sights are apart, the less pronounced errors in aiming will be.
I don't hunt deer myself, so I can't say how far I'd try that with peep sights. But just shooting targets using a stump as a rest, I get about 2" groups using budget .357 ammo (Magtech) in my Marlin 1894c with Skinner aperture sights. I consider myself a casual recreational shooter, and I have no doubt that better shooters with more carefully selected ammo would tighten that group up a lot.
As mentioned above, most aperture sights are either positioned at the back of the receiver or on the tang of the stock.
Tang-mounted sights offer the longest sight radius and may boost visual acuity most, since they are closest to the eye. If absolute precision is your goal, a tang sight is probably best. Potential drawbacks are that tang sights often require you to drill into your stock for mounting, and, depending on how you hold the stock, they may be in the way of the thumb on your shooting hand. As far as brands, Lyman and Marbles seem most common for tang sights.
Compared to receiver-mounted sights, tang sights tend to be larger, more expensive, and may be less bump/snag resistant out in the field than receiver or barrel mounted sights.
Receiver sights can typically be mounted using the same screw holes you'd use for a scope mount and still increase sight radius an extra 6" or so. Since my guns are more for plinking and small game hunting than serious target shooting, I prefer receiver mounted peeps. Among receiver sights, Williams, Skinner, Lyman, and XS are ones that come to mind. I'm most familiar with
Skinner sights. A couple things I like about the Skinner sights are that they seem designed to blend into the lines of a rifle aesthetically, rather than look like an obvious add-on. Unlike some options, you can easily swap different size apertures for different needs and lighting conditions... bigger apertures for closer, faster aiming or dark conditions; smaller apertures for more precision or bright conditions. Not putting an aperture in at all leaves you with a ghost ring, which offers very fast aiming at the expense of precision (probably most useful for home defense or hunting dangerous game at close range).
As for barrel-mounted peeps, I own a
Skinner barrel-mounted peep and have tried a Marble's barrel-mounted peep. Both fit into a standard rear sight dovetail. I really like the sight picture of the Skinner (using it as a ghost ring, since it's farther from my eye and therefore looks much smaller), and it does provide benefit #2 listed above, but it doesn't provide benefits #1 or 3. I didn't like the Marble's sight picture.
Another consideration is the front sight. Many peep sights sit higher than factory rear sights, and so a taller front sight may be needed. I don't think I needed a taller front sight with the Skinner barrel-mounted sight or their low-profile receiver sights. However, even if I didn't
have to change them, don't like the standard front bead of most guns. I tried fiber optic beads and didn't care for those a lot either. In bright light, they flared too much for my eyes, and precision went down. I found I generally prefer a flat-topped post or blade over a round bead with aperture sights. A couple of my guns have these
XS front blades. I know the description says "AR-15" but they fit in all standard front sight dovetails. I prefer these over other blades I've tried, because the white on black gives me good visibility in different lighting conditions against both light and dark backgrounds...and no flaring like a fiber optic. (the reason I don't use XS rear sights is that they offer only ghost rings, and I like the option of smaller apertures)