Problem: there may be several Inventory floaters simultaneously open (or none at all) and for each floater, there are as many search filters as there are tabs (full inventory, recent items, worn items, last open folder...)... Not sure how you'd like this implemented.
Search results ? But how ? The inventory search filter updates in real time with the inventory contents and search filter input line editing while Lua scripts are restricted to callbacks that execute in only a few dozens of thousands of instructions maximum (or else they timeout)...
Not doable from Lua alone: this would involve modifying the search feature for the inventory, but the latter is already very time consuming despite the presence of a caching scheme. Adding regexp support to it would make it super-laggy.
Same remark as above: it's beyond Lua scripting scope and belongs to the implementation of the Inventory floater itself.
Same as above. Note that you can use this trick for an equivalent result: close and re-open the folder you want to see in full, in the Worn tab, then switch to the "Last open" tab: the folder will be listed there (and you may "Lock" it too, with the corresponding check box).
Again, an inventory floater implementation detail that won't belong to Lua scripting (even if, once implemented, it could be called via a special Lua command).
Doable...
I could possibly implement a function that lists the contents of an inventory folder, returning the item names into a Lua table. Parsing for permissions would then just be a matter of checking for "(no transfer)", "(no copy)", "(no modify)" sub-strings in each item name...