My first problem is that I have no apps on my iPod desktop. The second is that my battery died, and despite being able to charge it, it is unable to restart back up.

Question

My first problem is that I have no apps on my iPod desktop. After using my computer to install something onto my iPod 5, it somehow disconnected. When i restarted the iPod, some apps (Safari, FaceTime, Game Center, App Store, iTunes Store, Camera, and some others disappeared. Apps like Flashlight and Jetpack Joyride which I had previousyl downloaded from the App Store were still there. I wanted to get the aforementioned apps back, so i decided to back-up my iPod to iTunes. The downloaded apps weren't able to be backed up ("authentication problems") but some apps like Tips, Video, iBooks, Reminders, Stocks, Settings, Voice Memos, were backed up. I then proceeded to factory reset the iPod, deleting everything and starting it as a new iPod. Once I finished this, I looked my desktop to see everything was gone, absolutely everything. I double-pressed the home button, showing Settings was open, but that's it. I opened settings and factory reset it again, this time choosing to restore from a backup. After doing this, I found the iPod looked the same as the last time i reset it - blank. If I scroll down the middle of the screen to bring up a search bar, I can find only the apps I backed up (Tips, Video, iBooks, Reminders, etc.) That is the first MAJOR problem. I was getting really frustrated, so I opened up iBooks and discovered I can download a could free books from the store, which was essentially the only thing I can use the iPod for. There were no videos in the Video app, and pressing the "Go to Store" button did absolutely nothing. No lag, it just ignored my touch. Once i found a few books, I decided to start reading. Eventually, i brought this iPod somewhere to read it, and didn't bring the charger with me. The battery died, and when i got home, tried charging it.
Which brings me to the second problem. A minute or so after I plugged in the charger into the dead iPod, the Apple logo came on. But the iPod never started up. The logo stayed there, getting dimmer after a minute, but no vanishing even after fully charging or pulling out the charging cord. It didn't leave even after hours, but by then the battery died again. If i press and hold the power button and the home button at the same time, the Apple logo disappears. But I can't get into recovery mode utilizing the volume buttons. If i hold the power button when the screen is blank, the Apple logo comes back. But the iPod cannot start up.
After everything was wiped, i was pretty angry. But I still had books to read, even if i couldn't do much else. Then the battery died, and it really wrecked me up. I feel so mad right now, although this actually happened about 4 days ago. The iPod being wiped happened about 3 days after the Christmas holidays began. This iPod also was not brand new, I bought it off a friend who had it for over a year, although it was in perfect condition (Obviously out of warranty.) I desperately need a solution to this HUGE problem. Due to personal reasons, I can't go to a Genius Bar or go to an Apple retailer in person, but communicating by E-mail is an option, as is instant messaging.

Answer 1

Try:                                               
- iOS: Not responding or does not turn on           
- Also try DFU mode after try recovery mode
How to put iPod touch / iPhone into DFU mode « Karthik's scribblings
- If not successful and you can't fully turn the iOS device fully off, let the battery fully drain. After charging for an least an hour try the above again.
- Try another cable                     
- Try on another computer                                                       
- If still not successful that usually indicates a hardware problem and an appointment at the Genius Bar of an Apple store is in order.
Apple Retail Store - Genius Bar
The missing apps could have been done by setting the Restrictions that can hid those apps. If the backup was made with those retrictions set the the Restrictions are also restored.
Thus, if you get it to work restore to factory settings/new iPod, not from backup                               
You can redownload most iTunes purchases by:        
  Downloading past purchases from the App Store, iBookstore, and iTunes Store