First off, let me say that, as a left-handed number pad user, I was very pleased to see the keybinding issues being addressed in the new patch. I've gone through and tested everything I could think of as it pertains to my own typical keybinding layout. Here is a summary of what I have found:
WASD
Remapping the WASD keys works properly for movement, Pip-boy menu navigation, and Transfer menu navigation. It does not work for lockpicking--this remain hard-coded to WASD. It also does not work in VATS for cycling between enemies and body parts.
E
Remapping the Activate key works properly for almost all instances I tested where the game expects 'E' (Activate, Grab, entering and exiting Power Armor, and Accept in Transfer and Crafting menus). When used for Accept in menus, the prompt still shows 'E', but the remapped key works, and 'E' does not. Remapping the Activate key does not work in Workshop Mode. Build remains hard-coded to 'E'.
TAB
Remapping the Pip-boy key works for opening and closing the Pip-boy, but other functions of the TAB key remain hard-coded to TAB. These include Cancel or Exit in menus such as Transfer, Crafting, Inspecting items in the Pip-boy, the Perk Chart, and cancelling VATS. VATS is the one particular case where this is more than a minor annoyance, since when you're in VATS you are in a non-paused state and in combat--not the best time to take your hands off your normal controls. TAB also works (along with the remapped key) for closing the Pip-boy.
R
Remapping R works for reload and ready/holster, as well as opening the Transfer menu. It does not work within menus. 'R' remains hard-coded for functions such as Take All in Transfer menus, scrapping items in the Crafting menu and Workshop Mode, and dropping items from the Inventory menu. It would be nice to see menu and Workshop functions remap since it is used a lot, including out of menus for Workshop Mode.
Q
VATS activation remaps properly, but menu functions of Q remain hard-coded, such as assigning items to the Favorites menu in the Pip-boy, and tagging for search in Workshop Mode. It also does not work for selecting a target in VATS.
X
The Auto-Move function can be remapped from 'X', but menu functions (like Inspect in the Pip-boy) do not go with it (not critical).
T, Z
These keys are used for Wait and menu functions like Transfer All Junk and Sort, but they are not remappable.
Alt, Ctrl
These keys can be remapped for their primary functions (Crouch, Bash/Grenade), but their menu functions (next/previous rank in the Perk Chart) do not remap (it's not really critical that they do, honestly). Also, here's a big one I just found the hard way--the remapped key for Bash/Grenade does not work for Hold Breath when sniping. When I try it, I continue to sway, and throw a grenade as soon as I release either the Bash key or the Aim button. Left-Alt is hard-coded to Hold Breath.
Arrow Keys
These keys can be mapped to functions like movement, but they remain hard-coded for menu navigation in Workshop Mode. This is obviously a problem for people who use them for movement, since they can't move while building settlements. Workshop Mode seems to have in fact retained all of its hard-coded keys. E, Q, R, TAB, mouse buttons, and arrow keys will not work for any remapped functions in this mode.
Anyway, that's what I've found so far in my testing. I'll add anything else that I come across as I play the game. Also, if anyone else finds other remapping functions that don't work, post it and I'll add it to the list in the OP. Thanks to schlumi for pointing out the broken mapping in the VATS interface.
The TAB stuff is probably the biggest remaining annoyance for me personally at this point, since there is a lot of going into and out of menus. Workshop Mode is also quite awkward, as is lock-picking, and sniping is pretty much broken if I can't reach the Hold Breath key.
In a lot of cases, the prompts at the bottom of the menu screens can be mouse-clicked, so as long as the game is in a paused state (most menus other than VATS), the remaining hard-coded keys are minor inconveniences (for me personally). Still, it would be nice to see them addressed when the patch goes fully live.