Sycamore or beech should be fine as they are often used for food items. Someone I was in discussion with about baby rattles said that trading standards had told them that they had to be fruit woods but there's no actual regulation that states this, just that it should be fit for purpose. One person commented that sycamore can be brittle so should be avoided but I've not had any problems with it in the tests that I've run.
I'm sure it will be, but also make sure that the ring is wider than the EU toy regulation size. I can't remember the size but there was a discussion about rattles recently. You may not be selling them or a business giving them away for free, but you might as well stick to the regulations in an attempt not to choke anyone's child.
For a teether I suspect that any finish you apply won't be in place for long unless you use a tough plastic coat. I'd probably just burning them with shavings and explain to the parents that they will dull. For my rattles I use Rustins Danish oil and it survives mouths fairly well.