to fit mid size grips, does anything have to be done to the shaft or is all the padding within the grip?

I'd like to get some clubs with standard grips online but have them regripped at a store, which works out cheaper overall.

If I am buying clubs from my pro he will normally include a regrip free to the type of grip I like.

Have you asked the online retailer if they will do the same or add a layer of tape to the grips.

In answer to your first question it does depend upon the actual grip being fitted but as general rule the hole will be the same size but have extra thickness to the outside layer

Generally it is only the exterior diameter which is larger (and thus the grip will weight a few grams more due the extra material). Almost all mens grips come at a standard 0.6inch core size (inner dia).
For example the core size of a standard Tour Velvet is the same as the midsize and even the jumbo, but the weights are 51.5, 53.5 and 61grams.

not normally however you can fit the same grip onto a 580 600 and 605 shaft raw and they all feel different.
So make sure you know the bore size you are fitting onto then you will always get the correct thickness.

For example, I'm thinking of buying a wood, which comes with Cobra Tour Velvet 50D grip.
Would I be able to fit a Golf Pride midsize grip onto it afterwards?

The online retailer won't do this as it's a clearance sale.

yup no problem at all