Should my train be carried?

You shouldn't have a problem with grass stains, since your dress will rub lightly (and one assumes they won't mow immediately ahead of you walking down the aisle). Depending on the lace, you may have some small snagging, but again, your dress will rub lightly, you *shouldn't* have big tears (unless those stairs have big cracks and that kind of thing, then you take more risk). FWIW, I think you may be overthinking it - but if you have concerns maybe ask someone at the venue about it and what brides have done in the past?

I wouldn't worry too much about it the grass stains if any will be under the dress

I would not have her hold it. Walking with it flowing behind you will be just fine. I don't think you will have any issues with grass stains or snagging.

First off, your location is so beautiful! I myself am having an outdoor ceremony with a train on my gown and am walking down a similar stone staircase with greenery around it. I am opting not to have my train carried for the simple fact that I want it to be special for my dad and I when it's our turn to walk. Just the two of us. I recommend if you're biggest concern is the staining or dirt getting on the dress, which is a fair concern, I suggest maybe finding a similar fabric and take a test run? Rub the test fabric on the stairs and on the greenery around the stairs so you can see what may or may not happen? Again, this is just a suggestion. Congratulations on your engagement and Goodluck on your big day!