How the price is formed, what exactly in Neo4j database is taken into account? I mean the database size or number of nodes/relationships/properties plus number of executed queries… etc. ?
Aura Professional pricing is simple, and the details of that pricing are included in the create database flow. In the page you linked, there is a "View Pricing Table" link that summarises this.
In short you pay an hourly charge for any hour, or part thereof, that you have a database up and running in Aura. You need to delete the database to stop being charged.
The price varies based on the size of the database (by Memory, CPU and Storage allocated to the database instance)
Within Aura Professional databases you can run as many queries and fit as many nodes and relationships as the storage space allows.
Each graph is different, so there is no strict guidance as to how many nodes or relationships each size database can hold.