Am I allowed to keep using the products after next year?
Yes you have paid for them, the year to year renewal of subscription is to keep you entitled to updates for all Object Desktop apps.
How do I sign into the Stardock Central application if my subscription is expired?
Your login info will still work, all apps will still be there, you will just not be able to update after subby expires.
You can and should archive all the apps using SDC (Stardock Central), right click any app, Archive app, there are a few options.
The archives are stored in C:\Program Files\Stardock\SDCentral\BACKUP.
I browse to folder, deleted all. Then in SDC selected to download and archive all. If you do not renew you should do this just before subby expires. Then you can copy the archives to a disc so you always have them when/if needed to reinstall apps.