While many older "screwdriver" and manual focus lenses can be used to good effect on a D5100, that's less true for entry-level lenses like your 28-80mm and 100-300mm.
The 28-80mm came in two different versions. The first version had a very thin, hard plastic ring for manual focus right next to the filter ring. The second has a slightly wider focus ring a bit further down the barrel, and also has slightly better optics. Both are lightweight and overlap the focal ranges of your two DX zooms. Either can produce nice shots when stopped down a bit. For some shoots, using a 28-80mm may allow you to avoid changing lenses at a critical time, but I think you'll find that advantage is outweighed by the autofocus and VR features of your new lenses.
The Sigma has less range and is slower at all focal lengths than your 55-300mm. It's shorter, lighter, and uses smaller filters, but for a DX camera offers little else. For much of its range, you may find that focus confirmation ("green dot") is unreliable except in very bright light. I've never used the lens, but I can't imagine it outperforming a 55-300mm.
Both of these lenses typically sell for about $40-$60 in the used market. The 28-80mm will provide autofocus operation with most higher end Nikon dSLR bodies, but I'm not sure about the Sigma.
"There is no real magic in photography, just the sloppy intersection of physics and art." — Kirk Tuck