Is it possible to play Skyrim using the combination of a controller and keyboard? I like to play Skyrim using the Logitech controller, but with the additional mods I have installed I need more hotkeys than there are buttons on the controller. I'd like to use the controller for the more common and straightforward tasks, and have a wireless keyboard nearby to use for hotkeys. Does the keyboard input still respond while using a controller?

Thanks

With some mods Yes. if mods requires SKSE then keyboard input will work for hotkeys while playing with controller i have tested this myself mods like IHUD, Predator vision, Better vampires all works with controller

I can second this.
Although some mods won't allow you bind a function to a key which already has a function when using mouse/keyboard. (e.g. I is initially assigned to open the inventory when using mouse/keyboard, and so some mods won't allow you set it as a hotkey, however others will).