The battery voltage is fine (3.5-4.2 is normal).
It sounds like you've already done a great deal of troubleshooting work -- the "Apple Logo for 15 seconds" bit means, more or less, that the bootloader finished successfully and transferred control over to iOS, which then panicked due to some detected problem. Unfortunately, that doesn't really give us any more info to work with, but that's why you're seeing that symptom.
The fact that your battery voltage is reasonable, yet you get no visible result when not plugged into anything is very strange to me; I've never heard of such a thing.
It sounds like there's a corrupted install of something on there (which is why you get the Apple logo, briefly), and the hardware is in such bad shape that it won't let you reflash it (error 1601 means you are stuck in DFU mode). As Tri Emil Alim noted above, you should try another computer to be sure, but I doubt you will see a difference. You've done all you can here; your logic board is dead.
Not the answer you wanted to hear, but you did everything I would have done to test it. :)