Export only selected and reordered columns to Excel

I think the best way is to use the "Refactor fields" tool from the processing toolbox.

This tool let you reorder field, drop field, rename field and even using expression to change field value. You can then save the result in many different format including .xlsx

From the tool help :

Refactor fields :

This algorithm allows editing the structure of the attributes table of a vector layer. Fields can be modified in their type and name, using a fields mapping. The original layer is not modified. A new layer is generated, which contains a modified attribute table, according to the provided fields mapping.