If some food was harvested and sold and this was done on shmita year, and this harvest is later found or left in ones possetion,without having purchased it, is it permissible or desirable to be eaten since it has the kedusjat shviit?
In the case where the food grew by itself and was then harvested or sold in a forbidden manner, it does not change the status of the food, which is Kodshim like you say. It needs to be treated as such.
And yes, it should be eaten with the kevana of kedushat shviit. (We have been doing a series on this recently with a variety of sources).