This type of roof covering called

The most important considerations are preparations and safety.

Make sure to fully evaluate the condition of the rafters at the top of the wall interface point. If the top log(s) are as severely rotted as you say the rafters are likely to be rotted at that point too. If they are rotted then you would not want to be jacking from the outside ends of the rafter tails. It may require jacking from the inside of the wall perimeter. In the end rotted rafters may cause you to have to deconstruct the roof structure so that be necessary rafters can also be replaced one by one and then re-assemble the structure.

You would need to consider adding some temporary cross ties as close to the bottom end of each rafter as possible. This will help to keep the rafters from spreading when you start the lifting process.

There will also need to be bracing added outside the walls from the orange pieces down at an angle to the ground. These braces need to be located both parallel to the rafters and perpendicular to the rafters. These can be anchored to posts dug into the ground. The braces will help prevent the roof structure from shifting laterally as it is raised up off the walls. One way the braces can be constructed is with overlapping lengths of 2x lumber that are clamped together. As jacking proceeds these clamps are loosened and reset to allow for the overall lengthening as the roof goes up and shortening as the roof comes back down.

Before doing any jacking do not forget to ensure that the roof structure is completely free from the chimney. It would not go well if the chimney were to impeded the roof movement.

Lastly ensure that there is plenty of help available during the raising and lowering process.

All the shingles you have pictured need to be replaced. It is a matter of easing up the shingles far enough to gain access to the nails, pull them out without ripping the shingle(s) up, it will also require easing up the shingles above the damaged ones and pull THOSE nail out of the way too since they are going through the top of the next course down. Eventually the shingle will practically fall out and you can replace the shingle in reverse order.

If you are fortunate, the roof has not gotten hot enough to bond the shingles together. On 3 tab shingles the asphalt dabs that come with the shingle to bond the tabs of one row to another are easy to separate. The architectural shingles I replaced my 3 tab shingles with, the asphalt dabs were a different story. The three tab shingles popped up best while the roof was cool, while the architectural shingles needed a heat gun set to get the shingles really hot, but not so hot to melt them (the shingles), but enough to get the bond to release. I hope you do not have to go that route.

I use a ripping bar to ease the shingles up, I have not had any luck with any other style bar, unless I sharpened the edge a bit.