Hello LHoage, I have sat back on this post for a while and taken in much of what others have suggested and commented on. IMHO, E6010 rod is most commonly used in industrial settings for vaious applications where it's distinct characteristics are desired or possibly required. It is a DC rod and thus is not going to be used by the average home welder or hobbyist for his/her purposes unless they have DC capability on their machines.
E6011 rod possesses many of the same characteristics of the E6010 electrode with the main difference being it's ability to be operated on AC current or DC current. The general mechanicals and other characteristics of the rod are pretty much shared with E6010. Back to E6010 electrodes for a moment, there are too many different manufacturers of this AWS designated electrode to likely gain an accurate count of the exact number. Depending upon who you are speaking with at a particular moment you will receive many different and varied opinions on which one is best, has the best operator appeal, performs the best, and for what reason. The same could be said for types and manufactures of E6011 electrodes. If you take a look at Lincoln, Hobart, and other major electrode manufacturers you will probably find that each one of them makes 3,4, or more different types of E6010 electrodes. With slight changes in flux formulation a rod's characteristics might be altered just slightly to fit a particular application just a bit better than another type. Yet, they all posess the E6010 designations.
I know a lot of folks that prefer using E6011 electrodes in cases where a 60 series cellulosic electrode is designated. They feel that the slag comes off easier, there are less instances of arc-blow(whether being run on AC or DC current) and many other reasons and explanations for those feelings and beliefs. I am sure that there are others who swear differently. Personal opinion...... fortunately we are still allowed to have it.
Concerning the challenge that your instructor has charged you with. Most in industries where codes govern would likely not be doing the sort of challenge that you have been given. There are many reasons for this and many of the responses have covered this reasoning. Here again, you are in school, you are allowed to experiment and hopefully encouraged to do so. School is a place to learn, make mistakes where the penalties don't necessarily carry monetary responsibilities, be given explanations for do's and don'ts and why. Be challenged, the one you are working on now is certainly one of those. Your mention of having issues with heat control, electrode manipulation, arc length, rod angle, and any other number of other variables is certainly educating you on the finer points of welding with a cellulosic electrode.
The students in our program mainly work with E6010, E7018, E7024, E7014, E6013, E6011, and a number of other stainless, cast iron, hard facing, and additional specialty electrodes. The emphasis is on E6010 and E7018 as there are large families of electrodes of these two types that run similarly, thus being able to run these proficiently, a person can generally figure the others out reasonably quickly. The other electrodes that I have mentioned here are provided for a taste of how they might run slightly differently from the main ones.
What I am driving at here is to consider applying yourself as fully as possible and learning as much as you can while you are in school. School is the tip of the iceburg, but it's a good starting point. When you go out into industry and get a job is when the real learning will begin taking place. Remain teachable and strive to learn something new every day. Good luck on your present and your future. Best regards, Allan