1. The GraphML specification does not specify how size, color, and labels have to be stored in GraphML. Thus everyone using GraphML is free to come up with their own way to store such information. Unfortunately, this means that yEd would have to know which application was used for generating your GraphML file and how that application stores size, color, labels, etc.
In short, only structural information can be imported from "foreign" GraphML files.
If you know how to use XSLT, you might be able to convert your GraphML files into "yEd GraphML files" - i.e. convert the additional information in your GraphML files into a format supported by yEd.
2. Unfortunately, yEd does not support macros or recoding user actions for repeating complex processes.
If you run the layout algorithms with the same options each time, you can reduce the required clicks for that step by turning off option "Show Modules Dialog" on tab "Modules" of "File" -> "Preferences".
3. yEd's main editor area always displays all nodes. However, yEd offers four context views, one of which displays only the successors of the currently selected nodes. The context view are by default located on the left between the overview and the structure view.