Lead makes a good roof, but finding leaks in one (or any flat roof) can be a real bear as the leak below may be a long way from the leak above.
If the folded seams are (as I suspect from your description) standing up, except where walked on and flattened, then yes, the walked-on parts would be prime suspects for leakage. Sometimes the the seams are all folded down and soldered, but standing seams are not usually also soldered (it's a fair bit of fiddly torch work.)
Look very carefully for cracks right next to the flattened seams. If you don't see any cracks, run a torch over the flattened area (trying to anneal the metal before reworking it) and carefully stand the seam back up - look for cracks on the lower side as well. If there are cracks you (or a lead roof specialist you hire) can clean the area and solder them.
If there is a wall above the flat roof area, also look very carefully at the flashing where the wall joins the roof - likewise look very carefully around any penetrations in the roof.
Some folks will no doubt tell you to reroof, since lead is "old-fashioned" and therefore also "old" - depending on the shape the roof is in this might be the right thing to do, but both lead and terne-plate (lead coated steel) roofs can last for hundreds of years, so it is well worth trying to fix it first, IMHO.
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.
Best Answer
Slope the surface of the roof to one corner - the black surround hides it.
The slope does not need to be much (0.5cm per metre or 1/4" per 3 feet roughly - my guess) but the roof should not have dips in it ie you should not get puddles after rain.
Then organise the downpipe as required.