Do you think it is a must to book the huts in advance this late in the season?
It always used to be easy to get in the huts without booking late in the season. Generally speaking organised tours were no operating after about 20 August. But the buses used to stop running some time about 28-31 August. Now they are running the buses later in the season, this forum is now getting a lot more questions about doing the hike in early September. So things might be changing.
is it easier to hike from L to T/Skogar or T/Skogar to L
S to N is, on balance more uphill with a net height gain of about 600m, but apart from that I can't think there is much difference. N to S is a lot more popular, but I wonder to what degree that is conditioned by naming the trail in that direction.