Personally,PostgreSQL. Both have their advantages and disadvantages, but my standard arguments in favor of Postgres are that it:
1. Enforces data types natively. I've shot myself in the foot before with MySQL and 'polluting' data.
2. JSONB support makes it easy to use a 'hybrid' schema if you want NoSQL-like behavior or a junk drawer to shove JSON into.
Regarding your conversion questions, it's common to have an app that communicates with several databases, but I don't know of any architecture that would allow ACID transactions in two engines simultaneously.
As far as converting data, it all depends on what your development preferences are. I work mostly in the Ruby ecosystem, and I can tell you that I've used ActiveRecord to a Postgres database, and exported the results into a SQLite file. I'm sure it would be possible to use an XML parser like Nokogiri to convert XML into a SQLite file as well. If you prefer another major programming language, the tools likely exist to accomplish the same thing.