pet rock?

You'd need to scrape algae and have some type of filtration to avoid a stinky mess.

Sure depending on what you want to accomplish. If you want coraline growth you would need a small light, and maybe a small heater and pump to circulate your water a bit. With nothing else in there water quality wouldnt deteriorate very quickly but an occasional small water change would be helpful and restore some of the elements the coraline and other growth would be using up.
You would also need to be carful with the water quality you use it might just turn into an algea covered rock very quickly

Since you wouldnt be feeding the rock and adding nutrients I dont think any extra filtration would be needed other then the sand bed and the live rock