I am aware that the cards themselves will continue to exist on Hive-Engine as tokens.

The cards are not tokens on Hive engine I don't think. They are tracked on the blockchain according to the Splinterlands custom JSON transaction, but the cards are not directly tradeable via Hive Engine. There are third party markets like Hive account@peakmonsters. But even these might depend on Splinterlands APIs to determine the state of card ownership.

Can the game still be played?

No the only place to play the game is at splinterlands.io, which is run by the company. Were they to shut down, perhaps they'd open source the game play code.

Can anyone run a server (is it open source?)

No the game play is not open source.

Can all the game assets still be used (card art, sound effects etc.) in the game? Are those assets copyrighted?

The assets are copyrighted, although Splinterlands has stated they are okay with third party use under certain conditions. It was in a Hive/Steem post, but can't find this yet.

Can new sets be created? Does the governance system allow for people other than the original Splinterlands company to create and officially sanction new sets?

I think they stated they are interested in third parties issuing cards, but I don't see how that would be possible currently. These cards wouldn't work in regular gameplay, but only in other games that decided to use Splinterland cards.