I'd definitely wear gloves when making the solution. it's the silver nitrate that especially does the staining I think, not the sts. either way, when I mixed mine, I wore gloves, and still got a few black dots on my hands that lasted for a few weeks. must've been a few pin-sized holes in the gloves or something.They say to use gloves only because the silver will stain your hands black.
but the finished sts solution I always handle without goves and never noticed any staining on my hands. including the undiluted stock solution.
as for hitting other plants, I'd worry about that. personally I use only a few drops per plant. since it's a hormone-thing it only needs a very low dose to work, so I'd think liquid dripping on other plants would easily turn them too.
I use drops(more convenient imo), but if you spray, I'd take the plant to be sprayed out of your growroom, spray it somewhere secluded with a plastic box behind it to catch the spray that passes, and only put it back after it's dripped out.