I know I don't feel dressed without my handbag (my DH wonders why I have to have it every time we step out, but I do). My mom was the same way. As I get older, it's even more essential that I have it on my arm or nearby.
As to Ipods, well, there are times when one wants to listen to music or podcasts or lessons or news privately. The Queen seems to have so little time to experience her own moods and inner life (for me, that's what music is about). One does not always wish to share one's moods with the chauffeur, the guards, etc - one may wish to block them out and use the Ipod as a noise canceling feature (I know I would; I've just started using one myself, and more and more, it serves that purpose - it's lovely).
I love that she carries the little trinkets in her purse, too. Her handbag seems to serve many functions, it centers her, contains small essentials for her person (as if anyone would want to have to ask someone else to hand them their lipstick!), and the signaling she does with it is fascinating. I think many people in public life (my job involves far too much public contact, actually) develop these signals, and those who know us well know these signals, it happens naturally but easily becomes a code.
My daughter uses one of those hook things as well, as she is somewhat germophobic (I'm getting there too), and so obviously, the Queen's placing of her bag on the floor is a sign that she's really had it with that engagement (and must be immediately rescued or else!) When I was younger, I thought gloves were a mere fashion statement or affectation, now I see their immense value if people actually want to touch one's hand.
She's one really well-prepared and put together lady. I do wish she'd expand her choice of handbags (but otoh, I've settled in on a couple of styles myself, and find it very difficult to change, and it would hardly be fitting to see Queen Elizabeth II with the latest Marc Jacobs or Fendi bag - can you imagine her with a spybag???)