If I close the mail app on iPhone 5S, can I still be notified of received e-mails?

I usually try to close all apps not in use from my phone to save more battery, but I always leave the mail app running in fear that if I close it, I won't be notified when I get new e-mails. Can I close it and still be notified or not? Also, same question for Snapchat. If I close out Snapchat, can I still be notified when people send me snaps?

I have a big problem with my iPhone?

Ok, so I'll start off with telling you the specs of the situation here:
-iPhone 4s JB evasi0n IOS 7.0.4 (Verizon)
-Bricked (Not a thing works)
-Back is off
-Power button does not work
-BTW, it isn't a normal brick, there isn't even an Apple logo that is showing, it's just off and stuck

Alright, so there, that's is everything. Does anyone know how I could resolve any of these problems?
I tried putting small paper widgets in-between the lock button and the other part of the phone. Also I've tried using a knife, and a sewing needle to get this button to work. I know that I have to hold the power button and the home button together for 15 seconds then wait for the screen to come up on iTunes for the back-up (which I already saved all my info with a back-up earlier today). All I need is the power button! Someone help me out please? That'd be awesome.

How to switch iphone 5 carrier?

My mom just got a samsung galaxy s5 and she gave me her iphone but it doesn't have service. I have a samsung galaxy ring from virgin mobile and Can I de activate my account or something and use the iphone with a different phone number and stuff.

What does this message mean on my IPhone?

Next Bill Cycle: 05/07/14
Data (MB): 118.19 of 300
Messaging: 732
Group Messaging: 1248 of unlimited.

Does this mean I've sent 732 texts and 1248 group messages?

Added (1). No, does this mean that sent like about 1900 messages?

I dropped my iphone in water - 1

It was a free Iphone 4. I want to get a new one. Will Apple make me pay or will I Get a replacement for free, because a brand new one is free too?

Why doesn't my iphone have service? - 3

My screen broke so I had to take my phone apart to put in a new screen. I put all the parts back together with the new screen, restored my phone with itunes and now my phone doesn't have service. It connected to service for less than a minute today and then went back to having no service. How do i get it to connect?