Re: Bokken handling
Indeed, ask your sensei for their preferred way.
However, if you want a general method, I would present the sword with a more-or-less outstretched arm, holding the end/bottom of the tsuka (so that the recipient can grasp closer to the tsuba to get a good grip). Make sure that the blade is pointing towards yourself; I'm sure there's something of a sign of respect in doing so, as well as the obvious safety factor.
That's for swords. However, if I was passing a bokken to someone in the middle of class so that they could have a go (and then the person after them and the person after them, etc.), I would have it laid out flat across the palms of my hands, with my thumbs just covering the top. Edge pointed towards myself, so they're not grasping a metaphorical three foot razor blade with their open hands. Accompany it with a bow and allow the recipient to take the bokken from you in a similar manner.