Day 1 - I think that the Philosophers Path is better visited during the cherry blossom season. I have been there several times and three times in the autumn and the koyo, although at some spots are colorful, is not a must see in the fall, Ginkakuji being the exception. Visit during the spring and the sakura season when the views are spectacular. Skip it and have more time for your other picks that day.
Day 2 - We like to go to Arashiyama early during the day at that time of year to try and avoid the loads of crowds. My advice is to do your walking tour in the morning. I like to arrive before 9:00 am and be leaving the area before 1:00pm. Doing the walking tour at 1:00pm at that time of year is what I would consider grounds for suicide
You will see why when you get there.
Day 3 - Again, get to Tofukuji right when it opens and before the flag carrying crowds arrive.....and they will at that time of year......trust me on this. Have lunch in one of the small home restaurants you will see as you are leaving the Fushimi Inari Shrine area out the back entrance. Fushimi Inari will have almost no one there at this time of year and will be a welcome respite to the crowds you will encounter elsewhere. You have a lot on your plate this day and Kiyomizudera in the afternoon at this time of the year only adds to the crowds and your length of stay there.
Day 4 - Perfect.....maybe start Dotombori around 7 ish
Day 5 - a rush but try to accomplish as much as you can and be sure to catch one of the express trains to Nara from Kyoto and not the local which will stop at every stop and really lengthen your commute to Nara.
Day 6 - Osaka Castle for sure and its adjacent gardens would be the perfect fit before carrying on to your late afternoon flight and give you a few more camera moments before you leave.

As for suggestions to eat.....looks like you will have to eat in the vicinity your are in according to your schedule. You won't lack in the quantity of places to eat....trust me

Aloha!

Day 2 - You'd be better to go to Nishiki Market earlier in the day...closes around 6:00 and nothing gets restocked as it sells out so it gets looking a bit bare later in the day. Arashiyama will be packed so you'll need to reserve the train (and boat if you are doing that) ahead of time or you won't get on without at least a lengthy wait.