They very much do pick at it and eat it, if they at all can. I am not sure how bad it necessarily is for them (assuming they have grit) but my feeling is that it's better to keep from finding out
IMO the best thing, if you want to caulk seams (and it is not like you HAVE to), is to put the caulk or glue in when you are assembling the joint in the first place, so it is filled with caulk or glue, then screw the pieces together and wipe off whatever amount squoozes out. The space is thus filled but there's nothing to peck at.
If you have really gappy crevices that you want to retrofit with caulk, my personal preference is to add a piece of trim (quarter round, or whatever you happen to have lying around that'll fit) and caulk well behind that before nailing it into the corner or across the seam, and strike off whatever squoozes out.
For retrofitting things that aren't real gappy, personally I prefer to sift a bit of food-grade DE into the crevice (I am not a major DE cultist like many here, but in crevices it will do no harm and quite possibly some good). Alternatively, you could just paint or polyurethane over the crevice so it isn't a crevice anymore.
One thing you probably do want to avoid, if using caulk, is the super-duper long-life exterior latex or mixed-content caulks. I have used two of them (one on the barn roof, one mistakenly in the chicken building) that REEKED of nasty solvent for some days despite lotsa airing out, and I am skeptical it's good for chicken lungs. Plain ol' silicone caulk is best, if you use caulk - all it releases when curing is acetic acid (the vinegar aroma you smell) which isn't that bad and isn't that long-lasting.
Good luck, have fun,
Pat