See my other reply here for the first question.
Regarding a search bar — that is not currently a supported feature in Full view but it’s on the to-do list.
Just a little background on all of this: most of the views in the plugin are ones I custom built as I was creating the plugin. Full view is different — it uses the open source FullCalendar library for rendering the calendar output. As such I do not have complete control over what it can or can’t do, but I am adding supplementary features in ongoing updates. The other views all support the search/filter feature but I have not had the opportunity to get it integrated with Full view yet. I’m working on it.