For champions like Alistar, Orianna, Lee Sin, Gragas etc can you cleanse out of their ultimate's/skills?
All displacement type spells (knock-back, knock-up, pulls, etc) cannot be prevented or ended by the use of Cleanse. Additionally, these type of spells are not affected by Tenacity. The only other form of Crowd Control that is not effected by tenacity or cleanse is Suppression, such as Warwick and Malzahar's ultimates.
However, it is worth noting that there are ways to prevent these types of spells, such as Sivir or Nocturn's shield, Morgana's Black shield, or Banshee's veil.
A lot of information about the varying effects and counters of Crowd Control types can be found here: http://leagueoflegends.wikia.com/wiki/Crowd_control
Yes, you can. However, this does not prevent the knock-up animation from completing. It only allows you to cast mobility abilities during the animation, like Flash or Vayne's Tumble, which allows you to escape the knock-up.