This app was mentioned in 8 comments, with an average of 3.38 upvotes
Personally, I hate doubletwist. iSyncr is bloat free (no music player or anything to go with it) and totally seamless. It syncs wirelessly too, which would require a $5 in-app purchase for doubletwist. Source: I used doubletwist until I found iSyncr.
Linkme: iSyncr
Did you use the free or paid version of Doubletwist? Paid version seems to support Airplay.
Worst case scenario, rooting the phone (especially if you just stay with stock ROM) really isn't that bad, though I totally understand the frustration that it's even required for a lot of functionality.
Edit: Have you tried AllConnect? There's no mention of root needed on the Play Store page.
If you've got your heart set on iTunes, you have a few options. doubleTwist has Android apps and Mac/Windows apps that make iTunes work with and sync Android, but the app isn't free. A free option is iSyncr, so that could work for you as well. Apple Music is also releasing Apple Music for Android, but that won't be out for a couple more months.
If you're ok with going all Google, check out Google Play Music as your music manager. A plus side of using this is that everything is in the cloud, so there is no need to sync anything. New users get a 30 day free trail!
Then pay?
There's also this https://play.google.com/store/apps/details?id=com.doubleTwist.androidPlayerProKey
For contacts, photos, and email, reference Google's own Switch to Android page for details and instructions on how to migrate things over to Google. The short version: your photos are probably best uploaded to Google Photos, your contacts will be imported into your Google account (into Google Contacts), and your existing email service most likely has an app on Android or has integration into GMail. As for apps, there isn't any tool I know of that is able to find equivalent apps cross platform; I think that will have to be done manually with searching the Play Store. If you need any help with finding an app that does something, let us know on here; we're pretty good with giving good app recommendations.
You have a few options when it comes to the music: I've heard good things about doubleTwist when it comes to having iTunes on Android (app link - $5), but you'll also need a desktop app to help with the syncing between platforms. iSyncr - Lite is another similar option. If you're open to a more "cloud" option, consider uploading your music files into Google Play Music (app link desktop link). Having your music in Google's cloud is not only free, but it also allows you to have access to your music on iOS, Android, or any web browser.
This may not be of immediate help, but in the coming months we'll be completely revamping AirSync and make it use the architecture of our CloudPlayer app.
If you want to keep an eye on future development work, consider joining our private beta group.
Note: I'm the doubleTwist co-founder :) I'll DM you an unlock code for CloudPlayer in case you want to try it for music playback.