Thanks for identifying a correct email address. Below is the excerpt of the payment letter with what I think is the address they provided to contact them if I had an issue. Is that an email address?
No, it is not.
And this caused your problems that you, or maybe Google Translator, did miss-interpret "DE-Mail" as mail address. It´s a system used also by our (Austrian) citizens/companies to interact with legal institutions like courts, district authorities, internal revenue, health insurance, etc..
In my country on need to be registered and you get for each transaction a code which you need to enter so that your personal involvement is guaranted/accepted. For such interactions you need to visit their websites and to log in with a handful datas.
Quote: [...] Communication between De-Mail and regular email addresses is not possible. [...]" Unquote
Source and more infos at: https://en.wikipedia.org/wiki/De-Mail
" But the billing aspect that has a declared deadline, came by a much slower method. I had no idea what bill amount was coming, by what method and with what deadline, so please allow some slack."
The text reads "payment within a month of notification".
You need to know that a one month timespan to pay a bill is very patient.
You also need to know that "notification" is NOT "within one month from sending the bill". It means "one month as soon as you´re aware of the bill". All my letters/files I got from Wast were sent with registered mail, and I assume that was also used in your case?
I don´t know the US postal laws but I know ours (this is important because you were absent as you told us:" the rest by me being on vacation when it arrived").
Now to the main thing: When someone is absent from his/her home he need to go to the post office and tell them about his eg. vacation. Normally they collect the normal post (leaflets, letters, postcards, etc) for little money. But more important is that registered post and documents from authorities [https://de.wikipedia.org/wiki/R%C3%BCckscheinbrief
] get sent back to them with the notice "absent". So that they´re required to sent it again after some time.
Are "you" not going to inform the postal office and the mailman throws an information (as requested because you´re not at home) in your letter box the deadline (30 days) starts from this moment/day. And there is no valid excuse later as eg. I was abroad because my father was ill, my employer did sent me sursprisingly to China, etc..
As said, this are our rules and laws, you need to know yours. They might be very simmilar.
However, this is only valid/important when you are planning to disagree with their costs (what is pure nonsense by this price, IMHO). Regarding your info about not knowing the billing costs. This fact they tell everyone in advance (or at least told, didn´t have contact with them since a few years). The costs are in every case depented onto the research success, and they are usually unknown before.
Hence, don´t worry, the transport time of the letter/bill is not included in the deadline. I also experienced the Wast- staff as really helpful and kind so if your payment is reaching them a day/a couple days later they want call the cops on you
If you need help with translations feel free to use the proper subforum here.
I hope this post is of some help for you.