Perhaps the following helps, you can type in any application that is supported by Flow, and it gives additional optioins. For example dropbox gives 1 option in the standard menu, but typing dropbox gives 2.
In addition I haven't found any way that you can really create a completely new "custom" trigger. Some "starting" points do give the option to add custom values, such as the Http.
For publishing a Flow, I found that after creating a flow and you click done, you get the option to publish this as well. But by the looks of it, only Microsoft published Flows are usable as a template.