I have used stain and polyurethane on countless projects and have never had either adversely affect my skin, other than a little stickiness, so don't worry about that. You can always use latex gloves as well if you are worried about it. Stripper is another matter altogether, it will burn your skin on contact so definitely wear gloves and make sure you have eye protection as well, it's pretty nasty stuff.
To refinish your grips I would get a wood stripper. They make different kinds, some will pull up the poly and the stain just by sitting on them, then you either scrape it off or wipe it off with some mineral spirits. That might pull most of your stain out. Then you can reapply or scrape the stain off the checkering with something like a very small chisel, screwdriver, or putty knife. You will have to be very careful to go with the checkering and not take any wood material off. If you can find something small enough with a 90 degree bevel on it (if that's the angle of your checkering) you might be able to fold some very fine grit sand paper around the angle and sand with that.
Also, I'm not sure I would put polyurethane on it, I think it would fill up your checkering and you wouldn't get the texture afterwards for your grip. I would put whatever stain you want on it and then finish it will oil, like linseed oil or something like it. You might need to reapply it occasionally, but it will finish nicely. It will be tedious, but if you do it right, it will look good. Good Luck!