Fallout 4 Power Armor mods

A frame can have any mix of power armor pieces from any type of model on it. You could have T60 arms with T51 legs and an X01 head and chest, for example. These can be swapped out at-will at a power armor station, or, if you're currently wearing the frame, in your pip-boy's apparel page.

Each piece can have mods added, like a VATS overlay for the helmet or a blood cleanser for the torso. Mods can only be attached to pieces from the same model. A T51 blood cleanser cannot go on a T60 torso. To change the mods on a piece of armor, you need to use a power armor crafting station.

You say you have some X01 Mk III mods. In order to use these, you'll need a frame. For each mod, the frame will need the associated X01 piece. If you have a Mk III leg mod, you'll need an X01 leg. These pieces cannot be crafted, but instead must be bought or found.

To attach mods to the pieces on the frame, you'll need to bring that frame to a power armor station. There, you can attach any mods you already have, provided that there is a matching piece on the frame. For each piece on the frame, you can also craft any mods for which you have the prerequisite perks and materials.

In short, a mod needs to go on the matching armor piece from the same model, but armor pieces can go on any frame with any mix of other pieces.

Yes to both of the questions you have asked. A power armor frame is universal, and can accept any power armor pieces.

(This answer is similar to DCShannon's)

While you can put different types of power armor on the same frame, you cannot put X-01 Power Armor mods on anything other than X-01 Power Armor. The same goes for all types of Power Armor. You also cannot place X-01 mods on a frame, you can only place them on the power armor. If you did not know, here's a little example of what mods are for.

1. I go over to my weapons workbench because I want to modify my pipe
   pistol.
2. I see a mod I like. It's called "Light Frame Receiver". I get the materials to make it, and I craft it.
3. I go to test out my new gun on some raiders.
4. Oh no! I don't like this receiver mod. I want the one I had before back!
5. I go back to the workbench.
6. The receiver I had before requires some materials I don't have anymore!
7. But with the mod, the game pretty much says "Hey, he's made that already once, I'll give him this mod so I can remember that there is still this one that he can attach to his gun if he wants it, without needing to craft a new one."
8. Yay! I've got my old pipe pistol back, as well as the mod for the light frame receiver, if I ever need it!

That example is the same as the Power Armor. You've already built the X-01 Mk.III once, so you can put it back on without needing to use up the resources. If you need more X-01 Power Armor to put those mods on, you can find some at military outposts. Have a good time exploring the wasteland, and watch out for deathclaws! ;)