Q: iPod 5 will not turn on or charge or go into recovery mode

It has probably finally failed, considering the condition you describe. I suggest replacing it.

Turn on without Home/Power Button

If you do, they will probably offer you an out of warranty replacement, since the home button and the screen are shot.

Out of warranty replacement is $99 for 16 GB 5G and $149 for the other 5Gs

except:

If your iPod is ineligible for warranty service, it may be eligible for out-of-warranty service. For example, an iPod that fails due to accidental damage or liquid damage is ineligible for warranty service but eligible for out-of-warranty service. However, certain damage is ineligible for out-of-warranty service, including catastrophic damage, such as the device separating into multiple pieces, and inoperability caused by unauthorized modifications.

No, an out of warranty replacement is a refurbished device to replace the device you currently have and would turn in to Apple. It will be at a reduced cost.