Battens and purlins are molding in the new building;

To my knowledge, these tests include simple nutrient media (e.g., AgarAgar). More professional are the mikrocount® duo from Schülke (Google!). Storage should be at 29 °C. At this temperature, yeasts and molds as well as bacteria grow optimally :) After 48 h to 72 h, you can see if nice fungal cultures etc. have grown!

Here is a picture for Post #13

I wouldn't worry too much about that. It was the same with us. Once everything is dry, the mold will be gone too. And eventually, it will definitely dry out if no moisture comes in.

Thank you very much for the feedback! We have now had the major issues addressed, but otherwise had everything done as the site manager wanted. As long as the mold does not continue to grow hidden or even penetrate into our cellulose insulation, hopefully everything will be fine.

The insulation is in the ceiling, which means you have a cold roof.
A simple test on a windy day,
go up through the roof hatch, have the roof hatch closed,
and then you should feel a draft at the top, then the roof is sufficiently ventilated and everything dries out.
If the air is still and smells damp or musty, create ventilation, e.g. with underlay ventilation elements.
And if you have always been good, your partner will let you down again.

... Have now had the bigger things treated, ...

How so?