Disclaimer: This is not a legal advice site. I am not a lawyer. For a definitive legal advice, hire one.

[...] in case a game's copyrights expire can I use its sprites for my own game and claim copyrights?

As far as I know, copyright laws around the world protect content for so long that any digital game that has ever been created is still protected.

If you think the copyrights have expired "because the company who made it was shut down", think again: the company's IPs have probably been transferred to another company, or a bank that seized their assets. Unless the company has explicitly stated that they give up copyright and made their assets available to public domain, you can't assume the copyright has expired.

Also you have to keep in mind something: even if the game copyrights have expired, if the characters are still used somewhere else, they are probably trademarked, which would still prevent you from using the sprites of the characters.

Unlike copyrights, which do have dates of expiration, company trademarks are valid for as long as the company uses those trademarked items commercially. [ref]

So it's not just a question of "copyright expiration".

In my particular case I'm interested in this game's sprites [...]

In this case, even if the copyright has expired, since it's a Nintendo DS exclusive, I would guess that you still can't use the data, because it would imply that you use the software or the hardware in a way not authorized by the EULA.

Secondly, how do I know a game's copyrights have expired?

You ask the last copyright holder :) Depending on the company, however, you'll probably have to hire a layer to ask them, because their legal departments won't waste their time with non-layers. (From a webinar with a game-industry lawyer, companies like EA will ask you "Are you a lawyer?" as their first question. I'll try and see if I can find the reference.)

It does not cost you anything just to try and talk with them. Depending on the company (and your company) you could work a deal or something.

Now, this is assuming that the copyright holder has not released their material to public domain.

I guess that if they said "We're done with the franchise, here are the assets, do what you want with it, good luck and have fun", you probably could use the sprites.

Disclaimer: This is not a legal advice site. I am not a lawyer. For a definitive legal advice, hire one.

[...] in case a game's copyrights expire can I use its sprites for my own game and claim copyrights?

As far as I know, copyright laws around the world protect content for so long that any digital game that has ever been created is still protected.

If you think the copyrights have expired "because the company who made it was shut down", think again: the company's IPs have probably been transferred to another company, or a bank that seized their assets. Unless the company has explicitly stated that they give up copyright and made their assets available to public domain, you can't assume the copyright has expired.

Also you have to keep in mind something: even if the game copyrights have expired, if the characters are still used somewhere else, they are probably trademarked, which would still prevent you from using the sprites of the characters.

Unlike copyrights, which do have dates of expiration, company trademarks are valid for as long as the company uses those trademarked items commercially. [ref]

So it's not just a question of "copyright expiration".

In my particular case I'm interested in this game's sprites [...]

In this case, even if the copyright has expired, since it's a Nintendo DS exclusive, I would guess that you still can't use the data, because it would imply that you use the software or the hardware in a way not authorized by the EULA.

Secondly, how do I know a game's copyrights have expired?

You ask the last copyright holder :) Depending on the company, however, you'll probably have to hire a layer to ask them, because their legal departments won't waste their time with non-layers. (From a webinar with a game-industry lawyer, companies like EA will ask you "Are you a lawyer?" as their first question. The source for this is here. The rest of the video is quite interesting.)

It does not cost you anything just to try and talk with them. Depending on the company (and your company) you could work a deal or something.

Now, this is assuming that the copyright holder has not released their material to public domain.

I guess that if they said "We're done with the franchise, here are the assets, do what you want with it, good luck and have fun", you probably could use the sprites.