1) If you set Quick Search to match from the beginning of the FileName, there doesn't appear to be a way to:
---> Use an asterix (*) wildcard to allow the search to match any characters in the beginning.
The QuickSearch popup window does not allow any wildcards.
You can in option specify if it should match from beginning of file or use substring matching.
( Explorer Panel Settings > Display (Tab) > Keyboard handling > Quick Search > "Search for text parts anywhere in the file/folder name"
2) There doesn't appear to be a way to activate Quick Search via a Command/Button.
You mean so that you can assign it to a button or another hotkey ? Hmm that might actually be useful, I will investigate and see if it is possible.