Often, yes, but this is highly dependent on the task -- and sometimes the sub-task. For example, when writing finding the correct word for an idea is motivating, but proofreading for spelling is a chore. I can name a good number of tasks that I do because I am internally motivated, including looking up definitions, washing my hair, organizing my bookshelves, making checklists and flowcharts, grocery shopping, gardening... but just as many (and probably more) that I don't do unless I really feel I need to because it is expected of me.

I think that part of this is that unmotivating tasks tend to stick in your mind as always present and demanding, but the things you enjoy doing are all too fleeting. It is also possible that you are unconsciously defining conscientiousness as "things you should do but don't want to", in which case by definition they will require an external force.

To me these are much the same thing. I am one with the situation. When I clearly see what is necessary, the action follows.

At least, that's the ideal, which I don't claim to always achieve.

I might not understand what you mean by conscientiousness. The things that most feel like things I must do are going to bed and getting up. Are they conscientiousness?

This is exactly how conscientiousness feels to me - not wanting to do something but doing so because it's the Correct Action For This Situation. Generally, this applies to things that don't give me a direct, immediate benefit to do, like cleaning up after myself in a common space.

I don't think so... probably less so than other behaviors, in fact.

In Freudian terms, conscientiousness is very straightforward: in involves putting the lid on your id and doing what the superego tells you.