As long as the painting is protected by copyright, you may not use any copy (reproduction) you possess or find publicly without permission. This is true even if you are the original painting's owner. You have ownership of the thing, not the right to replicate it. There is ownership, but there is no copyright. They are two separate things with two separate protections.
All that being said, yes, old paintings are generally royalty-free. If the artist has not specified otherwise, then you can reproduce and sell your work. Just make sure you get permission first!
For example, if I own a Vincent van Gogh painting, I can photograph it for my personal collection without asking permission from his family. However, just because I can do something doesn't mean I should. They could give you trouble for selling reproductions. Also keep in mind that some artists' estates may require payment. Some famous artists such as Pablo Picasso had strict rules about their art being reproduced. If you decide to go ahead and reproduce a painting, be sure to check with the copyright holder first before making any sales.
In general, yes. Unless you have permission from the artist or the iconic painting is in the public domain. (If you imitate the Mona Lisa, you will not be sued.) It is difficult to predict if someone will bother to litigate, although the copyright holder may choose to sue.
Famous paintings can also be used in advertisements, provided that you have their permission to do so. If they object, then you should not use their work. Consider how much money you would like to spend on advertising - it is a wise investment!
It is important to note that because these are copyrighted works, others may also want to use them. If this happens, there will be no royalties paid to you. The same thing applies to images of famous paintings - without the owner's permission, you could be liable for copyright infringement.
However, a great way to learn about art history is by looking at famous paintings. You can find out what other people thought was important in those years by just studying which paintings were popular. Knowing about artists' lives helps you understand their work better. For example, if you know that Raphael lived into his early 30s, you can guess that he might have produced some outstanding works but probably not.
Finally, remember that copyright law exists to promote creativity.
On own the copyright to the original painting or photograph, the artist does not need to file anything. Another artist cannot recreate that work for the intention of utilizing or selling it for profit. That would be profiting without permission from the original artist's work.
In order to be considered a copy vs. a recreation, there needs to be a very high degree of similarity between the two works. Even if you use the same materials, if you alter the way you mix colors or apply strokes to your canvas, then you're creating your own version of the painting. This is called "artistic expression" and is allowed.
It is not illegal to create your own versions of well-known paintings, but be aware that if you sell these paintings they may not be able to be sold as originals by the original artist. They could also be considered derivative works requiring you to obtain permission from the original artist or copyright holder to reproduce their work.
The best way to find out if you can't do something is to ask yourself whether or not you will be taken to court if you do it. If you can't give an answer to that question then you should avoid doing it.
In short, if the artist has been dead for 100 years (regardless of when the painting was made), the painting itself is in the public domain, but the image of the painting has its own copyright, i.e., if you use a photo of the Mona Lisa in your game, the person who took the photo actually has the copyright to that image. They can decide to license their photos out, but they could also choose not to.
Here's how long various types of intellectual property survive:
Artwork: Artist dies; artwork is 100 years old; no copyright infringement intended: Public domain. Copyright expires after 100 years; someone else may have copyrighted it before it goes into the public domain. Image used without permission works only against the original artist or their heirs. If an heir wants to challenge the copyright on an image they need to do so within 10 years or it will be assumed they are okay with the image being used in this way.
Images of artwork: Same as artworks, except that images of artwork don't receive any legal protection until they are fixed in a tangible medium (such as a print or computer file). At which point they enter the public domain together with the artwork they depict.
Music: Songwriter dies; music is 100 years old; no copyright infringement intended: Public domain.