1) Is it possibile to use Direct Query having a MySQL database local instance, without using Services in cloud?
* there are 2 different questions here;
- Direct Query is a method to control/model the data directly at the database. If one does not use Direct Query then instead the data is imported into PBI and then modeled inside PBI. So one does not need to use Direct Query at all but it can be a more efficient method in some cases. I believe Direct Query is only compatible with SQL Server but I'm not 100% certain on this point...
- one never has to use the Service component of PBI - that is always totally optional - in regard to linking to data sources...
2) Is it possibile to publish .pbix files in a local server?
- the term 'publish' is not universal - but I will say the answer is no - in that there is no publish feature to a private web server direct from the pbix file. Instead one would refer to the embedding where it is published to Service and then embedded into your web application.