Tiny bugs crawling about behind the glass in photo frames or picture frames or inside the back of the frame are very common. They like humidity.
You will find loads of photos online to help identify or rule out bedbugs.
Finding a single beetle in a room isn't necessarily an infestation.
Depending on the size it may have been a vine weevil which has flown or crawled into the house through a window, or crawled up an outside wall or come into the house via a flower pot (they like to lay their eggs in the compost in plant pots). The larvae feed on the plant's root then crawl out of the pots when they have matured. They particularly like geraniums and primulas.) They are not harmful to people or furnishings but will often kill the plant.
If the beetle was very small and black it may have been what is known as a "carpet beetle" (about 3mm) or a "varied carpet beetle" (which is brownish patterned) see: en.wikipedia.org/wiki/Varied_carpet_beetle
They can come into houses from the outside via cracks around windows and doors. I found some of these in a bedroom a couple of summers ago. Mine looked black or dark brown but under a magnifying glass they looked the ones in the link above.
There were several on one wall and also a few on the surface of a duvet, plus a few on the curtains. I picked them off and disposed of them. There were probably around a dozen over the course of a few days. I washed all the bedding and vacuumed thoroughly under the bed and the carpet. I have not found any more since then.
I don't think I would have thrown out a mattress because of a single beetle without having first identified what it was.
I hope you can find a solution to the mould. We live in a 50 year old house that is prone to mould and will probably be buying a humidifier. Where we live gets a lot of rain (we are a few miles from the sea, on a hill) but this last three or four years our winters and springs have been particularly wet.