I have just tested it myself: The volume is saved even if the power source is completely switched off in between (in my case, this is also a power strip with a switch). However, the speakers have to be switched on again each time with a simple "tap" on the cable remote control. Just turning the power back on is not enough.
The Bose are connected to the PC with a 3.5 mm jack plug. If the TV has such an output, then it is no problem. If the TV has a cinch plug for audio, then you would have to use a corresponding cable (cinch to jack).
You are probably right. The boxes say everything else but the power, and one part is Chinese.... I never turn it up more than 1/4 ! As a monitor, it's loud enough, and the quality is BOSE-like !!!!
If you don't want surround sound, the speakers are also great for gaming. I've had them for several years and am very satisfied. For shooters, e.g. PUBG, headphones are obligatory anyway.
Yes, you can. Check the manual page 15: https://assets.bose.com/content/dam/Bose_DAM/Web/consumer_electronics/global/products/speakers/companion_20_multimedia_speaker_system/pdf/344839_og_companion-20-multimedia-speaker-system_en.pdf