How to power an attiny from a solar panel?
You power budget looks ok. But the battery says:

"very important that you only use a LiIon/LiPoly 
constant-voltage/constant-current charger to recharge" 


and

"Never charge or use unattended."


The battery claims to have overvoltage protection, but it's not clear how that is achieved. It may overheat and catch fire....particulary as it says that 

"the batteries we sell do not have thermistors built in"


so-- don't burn your house down.