After Erase & Sync, Can I Restore From Backup?

In iTunes, when I go to iPhone>Summary and check "Manually manage music and videos" under options and press "Apply", I receive a confirmation pop-up explaining that my phone can only be synced with one iTunes Library at a time.
It asks if I want to erase and sync, and I understand that it deletes everything and replaces it with what is in my current iTunes Library.
My question is: Will the backup that I just created (both on iCloud and on my computer) restore my contacts/photos/app data etc. After I erase & sync or will that be lost as well?

My iPhone front camera is blurry?

The normal camera works fine but when I flip it it is blurry almost smudgelike and i can't take any pictures with it-when i press the button nothing happens. Then i have to restart the app so it would let me flip it back. I'm not sure if this is a hardware or software issue- i have tried rebooting the app and the ipod itself but nothing changes
it doesn't appear to be damaged on the outside, but i don't know
thanks for any help!

to get an image idea: go to this one
My iPhone front camera is blurry

basically it's like the image on the far left but it's a worse and i can't take pictures with it

My 5c battery is dying quicker than normal?

I got my iPhone 5c in June and it's been great, but this past week it's battery life drains the entire battery in about 4 hours, I don't text non stop because I'm not aloud to even have my phone at my job from 8am - 5pm then when I'm at home I'm with my family cooking and cleaning, the only apps I use is snapchat and Instagram and every once in awhile I use my Pinterest. I think it's because I used my friends cheap charger for a couple days, but maybe I'm wrong just wants some opinions, maybe I hit something in my settings or an app I use is bad.

Why won't it allow me to download apps to my iPhone 4s from the App Store?

Ok so a couple of days ago I purchased two game pack for $0.99. One day I wanted to download something and this thing popped up saying, "your payment method was declined. Please enter another payment method." so I put in all of my card information, but I didn't have any money in my account so I used my boyfriends visa debit card and it still said the same thing. He has money in his account. It gives you the option to use a visa, master card or an amex. I chose visa because I have a visa debit card from Canada Trust but this thing still won't allow me to download anything. Because of this stupid thing I had to reset my phone thinking it would fix the problem but it didn't. Now I'm stuck with absolutely nothing on my iPhone because the App Store won't allow me to download anything.

How much to Cancel my at&t contract?

I'm under my mother's phone plan. My contract was due to end in February of 2015, however this past June (2014) my iphone randomly broke. I went to at&t and they told me I would have to purchase a new phone. Now I guess I'm under a contract with them until 2016. I really don't want to be because as I said I'm under my mom's phone plan and I have to share data with 5 people. I want to go to Virgin mobile. How much in total would it be for me to buy out or cancel my at&t contract?

Is it possible to test/run android apps on an iphone?

So, I've been told by my professor that he wants us to test our apps on our mobile devices. Thing is, I got an iphone. Is it even possible to run android apps on an iphone? Or do I gotta run out and get me an android phone?

P.S. I got a macbook pro

Damaged iPhone can't make calls

I took my water damaged iPhone 4S into a store that claimed they'd fix it, and after two weeks they returned it saying nothing could be done, it was dead forever. I had some vacation pics I was determined to get so I took matters into my own hands and took the phone apart, only to find that the store hadn't even dried it out, let alone tried to fix it. Anyways, after a lot of messing around and learning about the internal hardware of my phone, I figured out that the dock connector was the only part damaged beyond repair. After replacing it, I worked a little magic on the other parts that were having issues (stabbed the flashlight, which somehow fixed it bc it was permanently on and now works as it should, disconnected and reconnected the speakers, fixing them, and screwed around with the lock button until it would stop locking whenever it wanted to, etc), and now I'm left with one problem: basically everything I pay for through Verizon (making and receiving calls, listening to voice mails, and texting SMS doesn't work, although I will occasionally connect to my service. I can still connect to Wi-Fi and use it like an iPod with texting enabled (iMessage), but was wondering if anyone knew how I could finish the repair and get my phone back up to par. Thanks.
P.S. I would just take it into Verizon, but the new management in my local store is really rude and I'd like to avoid that if at all possible. Also removed sim card, it's back in now though.

How to see a list of applications that I have already downloaded on my new iPhone?

I got a new iPhone because mine was dropped into a pool. Its hooched. When i bought my new one the guy from ATT said there were no found backups OR ICLOUD data. But when I download an app on my new iphone that I also had on my old phone it shows a cloud symbol instead of the download symbol. Is there a way to see the list of apps i had on my old iPhone because i'm having trouble remembering them all. I tried signing into iCloud on my mac but it doesn't show apps.

How to turn my iPhone 5 back on?

I had it charging last night and it was working just fine. This morning it was not responding to any buttons I would press or when I would unplug and plug it in to any cord or outlet. I tried pressing and holding down the home and sleep button, however, my sleep button does not work. Is there any other way to turn it on?