I had this same problem with both my imac and mac book pro using hotmail via mac mail with bt as my internet provider. Suddenly 4 months ago out of nowhere i could receive emails but could not send them. I scoured all the threads followed all the fixes none of them worked. However I just phoned apple support and fixed it in a matter of minutes. I am SO happy. This is what the technician advised me:
Go to your apple icon and scroll down to system preferences. Then dependant on what machine you have it is different for my imac and mac book pro click on either air port/wifi then the advanced tab in the bottom right hand corner. Then click on the DNS tab. Hit the + sign on the left hand side and type in 188.8.131.52 then hit the + sign again on the left hand side it will create a new line underneathe and type 184.108.40.206 then hit okay on the bottom right and then hit apply, again on the bottom right. Then send a test email to check but it should be fixed.
Had to share this. Its been driving me mad for months.