No you cannot change your category while registering for advanced as all the registered information you have given while registering for  jee main would be taken forward for advanced registration. In advance registration you would only get chance to change or add your mobile no.  or email id. If you have mentioned any category during jee mains registration then you will just need to again upload the category certificate during advanced registration and if you failed to upload valid category certificate then you would be taken under as General category if still you have given mains exam via mentioning your category.

And yes if you are taking drop next year then for sure mention your category while registering for jee mains.