Re: 1) "Safely Remove Hardware" assures you that nothing is being written to the drive and it is safe to remove the drive without losing or corrupting data. It does not power off the drive however.
The drive is powered through the USB cable. If you unplug the USB cable after getting the "Safe To Remove" message then the drive will power down.
RE: 2) I never install the software on these external drives. There is no need of it as far as I can tell. Windows can do everything you need. Keep in mind that "I never install this software", so if there is anything of benefit in there, I would not know.
If there is a Disk Imaging program, or a Hard Drive Diagnostics program on the drive I might safe those program installer files in a folder somewhere so I could install them later if necessary.
RE:3) Yes. Provided the "Safe To Remove" message has been displayed.
RE: 4) This I do not know. My external drives do not have power buttons.
This behavior could be by design (to prevent accidental disconnection, or something else.) Does it delay if the "Safe to Remove" has been displayed?
Not sure about dotnetframework35. I sounds similar to Microsoft program installer package " .NET Framework 3.5 " , which should be on your computer already through Windows Update.
I think you could safely delete that folder without any consequence.