One way to get good performance would be to write a SQL Server Integration Services (SSIS) package that connects to MySQL, truncates the relevant tables, and uses dataflows to copy the data from SQL Server tables into MySQL tables.
There isn't really a common backup format. You could have SQL Server write out a text file with a bunch of INSERT statements. Or you could write out plain text csv files for each table. Text conversions may become an issue. Excel is notorious for messing with your data based on what it thinks the data type should be.
Using SSIS is probably going to give you the cleanest inserts. It insists on clear data typing, code pages, etc, so you may find yourself writing derived column transformations in order to clean up small differences between SQL Server and MySQL. This may involve significant work, but the point is that it can be done in a consistent and predictable manner. And it should provide decent performance.