If I lost/have no access to the private key associated with a distribution certificate:
Can I just create a new one?
If I do, will it produce any problems during the submission process?
Will apps signed with the old certificate stop working?
Will I be able to update apps signed with the old certificate?
You can request a new Distribution Certificate on the iPhone Developer Program Portal and then create a new Provisioning Profile for App Store distribution.
This should not create any problems for the distribution process.
Apps uploaded with the old certificate will continue to work and you will be able to release updates for existing apps.
Other Answer 1:
Please refer to this article by Apple and it clearly mentioned there that
Important: Members of the Standard iOS Developer Program can be assured that replacing either your developer or distribution certificate will not affect any existing apps that you’ve published in the iOS App Store, nor will it affect your ability to update those apps.