Can I just read an ebook on the computer, WITHOUT downloading it?
In most cases, yes you can, although you will have more options (annotations, remembering pages, etc.) if you download.
Once I download an ebook to my computer, can I move it to some other computer?
Yes, as long as you are able to authorize the devices with your Adobe ID (so, download to your own computer, not a Library PC).
Will ebooks just disappear from my reader once it is time for them to be returned? (I do no have to worry about returning them?)
Yes, they will just disappear off your reader, you would not have to worry about returning them.
ProQuest Ebook Central
You can download ebooks that have print-to-PDF and chapter download capability on any device that supports .PDF file types – from Kindles and tablets to laptops and mobile devices. And you may transfer files between devices without support limitation.
eBook pages saved as a PDF can be transferred to, and read on eReader devices such as the Nook or Kindle. Consult your device's user manual for instructions on how to manually transfer files to the device.