Hello,
Every class will have a internal class number (KLAH-CLINT)
To check if material is assigned a class or not you can use the below logic,
Pass the material number to KSSK-OBJEK and get all the internal class numbers KSSK-OBJEK
Based on the internal class number you can get the class name from KLAH-CLINT
The characteristic values would be stored in AUSP, you can get them by passing AUSP-OBJEK = Material number
If you have multiple materials you can pass them at once to the tables export data into excel and then get the information you want.
For LSMW in background you can see the spool and logs generated in SM37 transaction.
Hope ths helps.
Best Regards,
Ameya Beri