What kinds of discipline will be acceptable when/if staff violate these new policies? Will we need to re-write our discipline policies?

Your policies mean nothing if you don't enforce them and you may to enforce them with disciplinary action. OSHA does not see that as retaliation, as long as you are consistently enforcing your policy. For example, if you are enforcing face coverings in one department you also have to enforce them in the other departments. Keep in mind, there is the possibility of difference in safety procedures if there is something so different about your workforce that you have come up with a different policy and procedure based on your risk assessment. The key factor here is that the employer must come up with policies and procedures, with a plan for discipline/enforcement as needed to carry out the plans consistently.

How do you suggest we keep track of temp checks and other "checks" if it's confidential?

One thing to note - avoid having a large group of employees stand in line for temperature check in the morning, try to stagger the arrival times to avoid a large group standing near each other. As the employer, you must have a confidential file, such as a medical file, that you keep away from your personnel file. You could keep the daily temperature checks in that filing system. You could also keep them electronically and only give permission to your admin users who have access to employee's medical information. These temperatures are HIPAA protected, so you do need to keep them separate from where other users could access them.