I bought an XBox One Controller to use with Fallout 4 on my gaming PC. So far, I have not used the keyboard and mouse at all, except for the debug console. Yes, the debug console is available even if using the controller. Took some getting used to, but now I really think it has made the game more enjoyable. No searching for the right key to press when death is imminent.

Does it work for your 360 controller?

Do you need a wireless adapter to play wirelessly

I'm using an xbox one controller as well. It's mapped pretty well. One little trick I didn't know is that you can actually zoom in on your pipboy so it takes up more of the screen by pressing the select button. For mouse/keyboard users you just need to Right click.

You don't need a wireless adapter unless you're dead-set on using the controller wirelessly. My sister uses the controller on my PC, and all we did was connect the charging cable to the controller, then plug the other end into a USB port.

I am using the wireless controller plus adapter package that Microsoft offers. I added the rechargeable battery kit. Windows 10 only. Lasts for my 12 hour gaming days, or can charge while gaming if I use the cable. Have not tried my Xbox controller; I don't think the adapter is compatible.
Side note: the charging indicator is built into the charging cable. Neat, if a bit weird.

Talking about the right click on pip boy, I've found that click-hold right mouse whilst dragging up and down will move the pip boy around. I'm not what function this serves, just thought I'd mention it.

I've recently got a controller and gave it a little go, I've been K&M for years so I found it rather awkward. I think I might give it another go as I keep hearing good things about the controller. In my brief time with it though I did find settlement management a lot easier to navigate with controller than K&M.

Interesting about the pipboy. It could be to center it on the screen or something. I might mess around with that later.

The only things I found that were easier with a keyboard and mouse was facial customization because dragging is more intuitive with a mouse than with a joystick and exchanging items between you and an NPC/container because you could simply click on their inventory rather than toggling it with the trigger.

Do yourself a favor and learn the keyboard + mouse

Else you'd better be playing a lot in 3rd person close combat or buff your companions because there is no aim assist on PC

Mouse + kb will allow you to be competitive in multiplayer fps as well

I do not have this problem

Like, ever

KB/M is more smooth for fps gameplay but hey to each his own