I have already submitted all my PhD applications (to US universities) and I did not write to professors before applying. Turns out this was not a good idea as in case of a couple of schools, I found out later that one of the professors I mentioned in my SoP have left the school. (I understand this makes me look careless and out of touch but their webpages were still up on the older schools' website and updated to list very recent publications)
Firstly, I was thinking of writing to the other professors in the schools who's work I am interested in but didn't mention in the statement in order to avoid crowding it with names. Will this help mitigate the damage caused?
Secondly, I was also thinking of writing to professors at other universities I applied to to ask whether they are accepting any students? I did not do this previously because some schools discourage writing to professors directly and the professors, who are very busy don't always reply (understandably)
Thirdly, since I'm thinking of doing this in mid-january, the professors will possibly be even busier with admissions work in addition to their research and teaching duties. Would writing this late be annoying? I don't want this to have any negative effect on my application.