Some suggestions for making things as fast as possible based on my own experiences:
-You can get to the rotten VERY fast.
Here is my video on how to get through the entire gutter in 2 minutes without fighting a single enemy. Not the only way as people told me when I posted it, but I only have a video of this route heh. You can use the silvercat ring to drop to the gutter, use this route through the gutter and run through Black Gulch and get from Majula to the Rotten in like 10 minutes.
-If you keep the Antiquated Key when starting NG++ (don't remember), skipping Ruin Sentinel is fun and optimal. You need to clear Heide anyway to unlock Harvest Valley, so going through No Man's Wharf to Bastille is fairly quick.
-Similarly, you can get through Earthen Peak very quickly. If you shoot the pots on the ceiling with iron arrows/heavy bolts during the Covetous Demon fight, he tries to eat the hollows that fall out and you can finish the fight without him attacking you at all. As always, abuse Jester Thomas for Mytha, that guy shows no mercy.
Just some quick things I remembered when I was in your position.