What is the cost associated with Azure Cosmos DB? Does the bill depend upon the database or containers we create? Also, does the bill depend on when we read/write data to containers or starts billing from the time we create Azure Cosmos DB? If that's the case, is there an option to disable/pause Azure Cosmos DB Account in Azure?
Pricing for Cosmos DB is described pretty nicely here: https://learn.microsoft.com/en-us/azure/cosmos-db/how-pricing-works. To summarize, you are charged for the capacity you reserved as well as for storage.
To answer your specific questions:
Does the bill depend upon the database or containers we create?
Yes. The moment you create a database or a container with throughput, that capacity is reserved for you so you start paying for that whether or not you use it. Please note that you can create a database without throughput and in that case you're not charged.
Also, does the bill depend on when we read/write data to containers or starts billing from the time we create Azure Cosmos DB?
Account creation is free. The billing starts when you create a database or a container in that account with throughput.
If that's the case, is there an option to disable/pause Azure Cosmos DB Account in Azure?
Unfortunately no. You will need to delete the resources to stop the billing.