When I have viewed documents at a Family History Center or FamilySearch Affiliate Library in the past I was able to view and download most documents for future access at home. I believe there are some reasonable restrictions on the number of images that can be downloaded, to prevent an entire film from being downloaded, however you are unlikely to reach this limit.
There are some documents such as the Derbyshire parish register collection which do not allow downloads for licensing reasons. I think this is more of an exception, personally I have not come across any others that cannot be downloaded. You can take a screen shot in these cases to save a portion of the image, if needed.
The copyright will depend on the source of the images; this is no different from documents downloaded from Ancestry, FindMyPast, or any other site. As a general rule it should not be a problem to keep document images for personal use however you should not be publishing them elsewhere (online or otherwise), without obtaining the appropriate permissions.