What is the most adviceable way to combine a staging and live database?
The way we do it is we take a copy of the live database when we begin development work, do the dev work on local/stage, get sign off... then manually add the new fields etc onto the live database. Otherwise, you're going to lose data.

I don't know of any way you could automagically merge two different databases.

One method to help speed up copying fields from one database to another (which is what you're doing) is to use the Field Manager plugin, which can export fields as JSON from your staging version, and you can import that JSON using Field Manager again on the live version. You'll still need to manually assign those fields to the associated sections.