Would you trust 13 year old tires?

For those who don't know, there is a 4 digit number stamped into the sidewall where the DOT information is. First 2 digits are the week and the last 2 are the year. Just bought a set with 1815 stamped on them - May 2015. I have heard that shops are not supposed to work on any tire 10 years and older.

i have not been able to find a shop willing to balance my N50s because of that

Correct on all accounts. The manufacturer recommendations are a little different by brand, but 10 years is the oldest that are supposed to be run and reputable shops won't typically do any repairs on tires that are 10 or more years old.

Also, the date stamping thing has been around for more than 10 years now, so, if a tire doesn't have a date code they won't service them either.