Does my iPod touch need service?

I have a 5th generation iPod touch 32GB and lately the device hasn't been performing as well as I expected. It currently runs iOS 8.4.1 and the problem(s) associated with this device, include the battery and Wi-Fi. Bluetooth works perfectly fine. After updating to iOS 8.4, the iPod touch failed to connect to my wireless network. It kept saying I had an "incorrect password", even though the password is valid and works on my other devices. So I tried connecting to another wireless network and it failed to establish a connection, even though the network was a public hotspot. The battery has been causing problems, too. I have to keep my iPod on the charger all the time, because it drains from 100% to 20% in a hour or less.

I tried everything to get the wireless to work and still nothing has worked. I had to restore it in iTunes via recovery mode, because it couldn't connect to Apple's servers on its own. I restored it from a previous backup from before iOS 8.4 and even to factory settings at least a dozen times and still, nothing has worked.

So my question is do I need service and since I don't have an Apple Store anywhere near me, what are my options?

The iPod touch has a month of telephone support left and eight months of complimentary service.