Disclaimer: I am not a lawyer, you are not my client.
Your rights regarding clippings found on Ancestry.com and similar content-providing sites are limited by Terms of Service first, copyright second; their Terms of Service may limit your rights even if the clipping is free of copyright.
Once you know the clipping exists, it may be relatively easy to locate the same on another site, but there you'll be bound by their Terms of Service...
There generally are multiple copyrights involved; there is the original content and publication, there is the photographing and digitisation of that content, and possibly some image manipulation to enhance the image (and recognise it as theirs...).
If you found an original paper in your attic, you would not have to worry about those Terms of Service, but would you'd still have to deal with copyright law, which is a topic unto itself. However, a quick answer to your second question; that a newspaper is defunct does not imply that all their content is in the public domain. You may need to track down the current copyright holder.
The Terms of Service do not only restrict, but probably allow limited usage as well.
Do not forget that you have fair use rights for copyrighted content.
The page that shows the clipping probably mentions the copyright status, current copyright holder or collection provider.
Articles elsewhere summarise copyright laws for various countries.