Technically no it doesn't level faster provided you use the same spell's you did the first time through. However depending on how much you've upgraded Magicka and what spell's you have acquired, you can level faster by casting the higher level spell's, provided you have the Magicka to cast them. I recommend the Bound weapon spell's as you can simply unequip the weapon and cast it again. Not sure if it still work's, casting soul trap on a dead body repeatedly will also level up Conjuration.
To level faster you can use the Mage stone (Mage skill's level 20% faster) or the Lover's stone (All skill's level 15% faster) although the Lover stone won't stack with Rested bonuses, unless you use the Aetherial Crown (Dawnguard Add-On).
Kind of, having heavily Smithed / Enchanted Armour or Weapon's and making them Legendary then taking on tougher enemie's level's them quicker because you're able to withstand Giant attack's for longer and do more damage. The Magic and Thief skill's are easier because most of them you don't have to be in Combat. Illusion and Alteration skill's are the easiest especially if you enchant equipment that allow's 0 magicka cost, there is a forum post regarding easy Skill's to Legendary.
If you've done the Dark Brotherhood quests, you get the option to upgrade the Dawnstar Sanctuary. If you purchase the torture chamber, there are four human prisoners inside that you can use to practice soul trap, sneak, and many other skills/spells.