Soar throat and wisdom tooth removal are two different things and are not interlinked.
Coughing and sneezing won't affect your tooth extraction site as it's already 5 days ( by this time the healing process is well organised) and yes u won't have dry socket because of coughing or sneezing.
Mouth opening may remaining reduced upto 15 days if the tooth was removed surgically.
It's good that u are still taking soft diet you may gradually shift to normal diet in a day or two.
Next Steps
Increase the no of times you do warm saline gargle to atleast 7-8 times a day for another 15 days ; this will be useful both ways for the extraction site as well as your cough issue.
You should visit a physician for the cough issue if not relieved and take necessary medications
Also you should visit your operating dentist to get an inspection done of the extraction site.
Health Tips
Follow the proper oral hygiene measures ( Brush twice daily+ warm saline rinses 7-8 times for atleast 15 days + Use Chlorhexidine mouthwash wash after each oral intake )