According to this Time article from 2013:
But the biggest obstacle to postal reform, by far, is the problem of funding Congressionally mandated pre-retiree health benefits.
Since 2006, the Post Office has been legally required to pre-fund health benefits for future retirees at a cost of around $5.5 billion a year. For the first time last year, it defaulted on its annual payment.
This sort of answers a few of your questions, though changes a few others. According to this, the problem at least started with pre-retiree health care costs. Specifically--the mandate that they be pre-funded starting in 2006. The article seems to indicate that these benefits were mandated by congress.
It also seems to state that the first defaulting of said obligation was in 2012, 6 years after the initial mandate.
As for retiree health care benefits, it appears that they are given as part of the FEHB:
The Postal Service is part of the Federal Government, and that means postal workers are federal employees. As a result, career postal workers are generally eligible to receive health coverage through the Federal Employees Health Benefits (FEHB) Program. Some noncareer postal employees can also receive health benefits through the program.
The FEHB was created in 1960.
So, to answer the specific questions:
What year did the USPS retiree healthcare benefits start?
It appears it was 1960 (unless the USPS somehow opted out initially, though I can't find anything specific in that regard)
Did it start in 2006 when Congress mandated funding?
No, it was already in existence, which led to the mandate, as congress felt that there were going to be future funding issues with it as up to that point, they were 'paying as they go'.
Was it under-funded from its inception until 2006?
They paid retiree benefits every year. It wasn't pre-funded.
Was the retiree healthcare benefits created by congress or by the USPS itself?
According to the above wikipedia link, it was a combination of The Executive and Legislative branches:
Congress modified the Executive Branch proposal