I have a workflow that is created from a default "Approval" template, and the basic startup conditions are not enough for my purposes. I also can't open this workflow in SP Designer (it doesn't even show as a workflow in Designer). So, is it possible to kick off this workflow from within another SharePoint Designer 2010 workflow? In this way, I can specify a condition that must be fulfilled in order to start the Approval workflow. I checked under the available actions and it doesn't look like one of the options.
Starting a workflow from a workflow is not there OOTB. You can extend the available actions either by writing your own or using something like the SPDActivities on codeples ( http://spdactivities.codeplex.com/ ).
there are other methods where you can utilize multiple lists/workflows to achieve similar results, but without really knowing what the "condition" is you are trying to check, can't say that will work for you.