In my experience there are only a few and mostly exotic cases where your software supports *both* MYSQL and MSSQL. As the latter is more enterprise-centric, you'll mostly see applications supporting MSSQL together with SYBASE or SQLPLUS, OR Mysql with Postgres.
The few cases where I have actually seen both listed for the same product, is generally when the product is ORM-based, meaning that the actual database is mapped through a compatibility layer. This usually also means that the developer doesn't use any exotic or dialect-specific features, so that you can safely pick any of the recommended options depending on your experience and environment.
If we are talking a new database setup without prior knowledge of any of the solutions and cost is more important than support, go with MariaDB (a Mysql fork), or Postgres if supported. Both are great databases for most use-cases and have an active community and lots of documentations/guides for all possible cases. If support matters and you are fine with either Windows or Linux, my personal preference would be a Linux distro that includes operational support for shipped applications (e.g. Redhat)
Usually it's not a great idea to access a database directly over the pond, SQL applications usually don't appreciate the incurred latency. It's usually better to run the application closer to the database server and stream/host the application from a localized server. You may stumble over read-replication on some documents, but they are by nature generally not synchronous which is a great way to destroy your data consistency.
Backups work about the same in all SQL Dialects, dump as an SQL Export through the integrated tools and then store preferably off-server.