I believe you are correct when you say it is a combination of asthma and allergies. The runny nose and teary eyes would indicate to me that he is having an allergic reaction to something, what only testing would be able to tell.
The other symptoms sounds like he is having a asthma attack. The shortness of breath, coughing and wheezing. He will need to see his doctor to confirm this, but that is were I would put my money.
Try and convince your boyfriend to make an appointment to see his doctor sooner than later.
He does appear to have asthma attacks that are triggered off by allergic reactions. this is very common with people who have asthma. Living in an environment with mould is no good. Mould produces spores that we cannot see, these are then breathed in and cause problems. Your boyfriend also seems to be allergic to the different pollens from the grasses, trees and other plants.
The doctor will be able to get him to do a lung function test to confirm whether he has asthma. Sometimes the patient is sent to the hospital to have a lung function test done. If asthma is confirmed, the doctor will prescribe your boyfriend with the appropriate asthma relievers for him to use.
As he does appear to have allergies, the doctor may also prescribe an antihistamine for him to use as well.
People with asthma do have different triggers and he will need to try and avoid these if he can.
For the mould in his property, you can buy chemicals to spray to the kill the mould. If you do use the spray, make sure that you wear a protective breathing mask to avoid breathing in the spray. Having good ventilation, opening windows and keeping humidity at a reasonable level will also help to keep moulds at bay, even though it may not prevent it completely.
Definitely go and have a check up. It sounds like asthma or some serious allergies and there are medications for that, so it should make life easier for your b/f and for you to not worry so much. Did you notice that fastened and irregular heart beat on occasions or more often as it may also be an indication of arrhythmia - again, it doesn't have to be serious.