The reason you can't adapt E-mount lenses to your A-mount camera is the same reason you can't adapt rangefinder lenses to a dSLR--the flange distance. The flange distance is the distance from the image plane that the lens is held by the lens mount. This distance is specific to each mount system, and must be maintained for a lens to focus through its full range to infinity. If the lens is held closer than this, it cannot achieve closer distances. If held farther away, it can't achieve longer ones (say 10' to infinity. Think macro extension tubes.) The mirrorless mounts tend to have flange distances about half those of SLR/SLT cameras, because they don't require a mirror (pellicle or otherwise) between the lens and the sensor/film.
While the Sony Alpha A-mount flange distance is identical to Canon's, the throat diameter is small enough that the lenses will have to be held farther away than with Canon EOS just to fit on the camera. The only two mounts you can adapt simply with rings and not with adapters with glass elements in them to achieve focus to infinity are M42 (smaller throat), and Leica-R.
The Fotodiox adapter rings for Contax/Yashica, Olympus OM, Pentax K, and Nikon F mounts, despite their having a deeper flange distance, all have glass elements in them, and that will typically degrade the performance of the adapted lens. The same will also be true for Canon FD/FL and Minolta MD/MC lenses (which have a shorter flange distance than Sony A). Whether it's enough of a quality hit to matter to you is a personal judgement call. But if you want to eliminate the glass element, you are going to have to use a lens mount kit from Leitax to achieve that goal, or figure out how to machine your own mounts to replace the ones on the lens, and at that point, whatever "bargain" you were hoping to obtain has mostly gone out the window.
You do also need to understand that when adapting manual lenses, you don't have any electronic communication between the lens and the body. You lose lens EXIF information. You may lose metering (or at least wide-open metering), you lose autofocus. You lose a lot of the automated modes on the camera that rely on adjusting the lens's aperture (e.g., full auto, shutter-priority, programmed auto, scene modes). You don't have any guides in your viewfinder to help you with fine-tuning manual focus, like old film-era SLRs did. This is a lot more of a pain than getting a native-mount autofocusing lens, and whatever you may save money-wise you pay back in inconvenience. Unless you have a love of vintage glass and a contrary stubborn nature to simply do a thing because you can, this may not be the path for you.
Frankly, your best bet for cheap vintage glass on an Alpha body is to simply find old Minolta AF mount lenses, which are, in fact, the same mount as Sony A-mount. The dyxum website is your best knowledgebase for this, and for adapting lenses to the Sony A-mount.