There’s no need for an air gap between the wood and insulation, in fact it could render the insulation ineffective by allowing cold air to circulate behind the insulation boards.
And the answer is yes to installing a house wrap as it would help to mitigate air leakage, log homes are not known to be particularly airtight. But as for where you install the house wrap, you can’t go really wrong either way, but you would be better to put it against the logs as it would probably be a more effective air barrier. Covering the Rockwool boards with a house wrap and then attaching it with strapping may lead to more holes.
I would suggest you apply an air barrier / weather barrier membrane directly to the logs, then the insulation directly to the membrane with no air space. Attach the insulation boards with vertical strapping (not horizontal) to allow for air flow. If you plan to go with board and batten cladding you would need horizontal strapping, but we suggest a double layer (or at least shimming the strapping out) to allow for a vertical flow of air for drying. That would be the only air space you need, right behind the cladding. Rockwool is vapour and air permeable, so it should have no problem allowing any incidental moisture to escape and it doesn’t need to be covered by a weather barrier.
When choosing an air barrier membrane - you can go with a typical house wrap weather barrier, but that wouldn’t be my first choice as any inconsistencies in the surface of the logs and chinking could leave it loose and at risk of larger holes forming when you attach the insulation. I think your best bet for airtightness would be a peel-and-stick membrane (with primer) as it could better follow the contour of the logs and chinking. That would likely reduce the number of holes in the air barrier you end up with, it would also be more obvious where the solid wood is when installing insulation, so you don’t accidentally drive the fasteners into the chinking.
That weather barrier suggestion is assuming logs that may be rounded on the tops and bottoms, which would have inconsistent joints and chinking. If the logs are squared on the top and bottom, a traditional house wrap would be less of a risk as you could chalk a line on it to ensure the fasteners go into solid wood.
As for cladding – of your two listed preferences, our vote is for the pre-painted wood. Vinyl is for sure the cheapest and it works well, however, it’s important to note the environmental impact of vinyl as a building material as it is responsible for 40% of global chlorine gas production. Here is our page on choosing cladding if you’d like, and following are some other pages that may be of value.