What is your stopping point... I assume A1 hell for the 3rd imbue quest?
For your #1, this ladder season (hc) I have been leveling my mules for socket quests (norm + NM). They are all barbs and I equip them with sanders gloves, sanders boots, stealth, and ravenclaw or bloodletter, with a crappy insight on merc. These are all items I find constantly while mf in nightmare. With chant I can get thru norm A5 in about 1 hour. I haven't run them thru NM yet but I have 3 at level 40 so far and it has been quick and easy. With better gear I'm sure you would not need chant.
I like barb because of access to increased speed, natural resistance, battle orders, war cry. Using concentration with 1pt berserk. For softcore another class might be better.
For your #2, I would also recommend watching someone like mrllamasc who does a ton of speed runs, he probably has the most efficient way to do with zero gear. Classes that immediately come to mind are holy fire pally, fire trappasin, fire druid.
I know this isn't exactly what you were asking for but hope this gives you some ideas
