Yes and yes...
Yes, you can add material, usually no more than a couple of inches a year, to the top of an existing bed...usually heavy in the compost/composted manure.
And yes, the different varieties will cross...but why worry about it unless you want to save seeds...it isn't gong to matter very much, because you are harvesting something that isn't being produced from seeds, but rather the perennial root. Yeah, any chance seedlings would likely be crossed, but proper cleanup of the bed would nearly eliminate those.
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