Push notifications are notifications sent directly to your device, I have a thread about them on BYH: https://www.backyardherds.com/threads/guide-how-to-enable-push-notifications.40125/
The site loads quickly for me. Can you try clearing your browser cache?
Just take some time to get used to it, it really isn't all that different from how it was before. We're upgrading BYH and BYC, but the current custom design on BYC will be ported to the new version, so it won't be the same as what is used on the smaller sites.
It's not that different from how it was before. Post bookmarks are new, so are additional post reactions other than likes if you hover over the like button. Also, you can use emojis typed from your phone keyboard now.