Apple TV - you are going to love it. My particular fav is airplay. I have it hooked up to Home Theater receiver and to my TV set. Good part - I can stream movies and music to those. Favorite part - when friends coming over and quiet music plays on the background, they do not realize that music changes as we talk, kind of like part of conversation. You literally control it on any Apple device and you choose what plays now, sound level and transition to another album or song seamlessly.
Photostream.
It does not depend on your backup in any form. As long as you have it "ON' on your phone and your Apple TV, every picture that you take, from now on, on your phone is going to be in photostream album of Apple TV and iphone.
Backup. Reason you need backup, imagine something happened to your phone and you lost all data. You will get another phone, but all your texts, contacts, pictures are gone. For that not to happen you need a backup.
What needs to happen now is you need to reduce size of the backup and go on good network. Your work is not good, common mistake. What they do at work, most of the time is they block secure ports, that would allow your phone to properly authenticate to icloud server. You need to get to fast open network. Apple store, Starbucks, MikeeD, Coffee shops, that kind of thing. You also may need to start slow.
How to start slow: two parts
1. First time you want to backup minimal amount, since we really don't know how large your backup is now. I have seen as big as 22 gigs, or as small as 1. In order to reduce size of the backup look in that article and for the first time you may need to turn off all of those.
2. Reduce something that is not on that page that you can control at the article above. What seem to be very common, biggest amount of space inside of the backup are imessage conversations. We tend to send pictures or small videos to our friends. So you sent one picture to 8 friends - takes 8 times amount of space in backup.
Clean those out if you can. Do it before you backup.
Good luck, I think I never typed that much.