Are you renting this house? If yes, any hope to move elsewhere with less problems?

It sounds quite serious and could impact your BF's health long term, depending on the nature of his cold and how long he has had it.

Would look into a dehumifier, ear plugs to sleep. A central heating would help but not possible for your place right now.

It sounds the house needs to be properly constructed/retrofitted with insulation, air pocket barrier for ventilation within the wall construction,etc. If you're pals with licensed engineer, architect they could give you simple explanation of the problem.

For certain, if you have wall to wall carpeting, have it removed.

We could not live in a place like that. My partner is allergic to mould. He can't go into even a greenhouse before he starts getting a huge headache, swollen eyes, etc. Heck, just having one indoor flowering plant to water is enough. He's afraid of mould from the soil indoors if we have too many indoor plants. I'm not kiddin'.