Math and Vector as root nodes

Math and Vector are the most used they should be in in the root node in the node menu.
You work faster with them available directly under node menu.

I just use search for everything. It seems the fastest, at least when you know what your looking for.

Hit space while in nodes display, does not display all nodes, and i find it works pretty bad instead.

use shift a and the top one is seach. I use that for everything I know the name of.

A direct shortcut for “search” would be cool.

Can we use space to directly access “search” ?

not really - in 2.79 it is a search for blender items - not necessarily for add on things.
in 2.8 the search is a second option from the space bar - unless they changed it again. - Animation nodes works with CTRL-a to go directly to search nodes but that is built into the add on.