1 Do I use mySQL server of a hosting provider in that case to run my database?
Yes. In your application you only change the JDBC connection URL and credentials.
There are other details about the level of service that you want for the database: security, backup, up time. But that depends on your hosting provider and your application needs.
2 Is it stored somewhere on the server?
Depends on how your hosting provider hosts the database. The usual approach is to have the web server in one machine and the database in another machine inside the VPN.
From the Hibernate configuration perspective, is just changing the JDBC url. But there are other quality attributes that will be affected by your provider infrastructure, and that depends on the level of service that you contract.
3 Should I declare somehow that data must be stored f.e. in a separate file on server?
Probably not. If your provider gives you a database service, what you choose is the level of service: storage, up-time... they take care of providing the infrastructure. And yes usually they do that using a separate machine for the database.
4 Am I going to loose data in case of server reboot? (As f.e. I do when I restart server on my local machine)
Depends on the kind of hosting that you are using. BTW Why you loose the data on reboot in your local machine? Probably you are re-creating the database each time (check your Hibernate usage). Because the main feature of any database is well... persistent storage :)
If you host your application in a virtual machine and you install MySQL in that VM... yes you are going to loose data on reboot. Because in this kind of hosting (like Amazon EC2) you host a VM for CPU execution, and all the disk data is transient. If you want persistent data you have to use a database located in another machine (this is done in this way for architectural reasons, and cloud providers like Amazon gives you also different storage services).
But if the database is provided, no.. a persistent database is the usual level of service that you should expect from a provider.