That is a good question. In pretty much all your cases, they could be interchanged, but would feel very slightly different.
One typically "starts off" on a journey - conveying a feeling of movement. One can also "start out" on a journey, but it feels more like the point in time when you began rather than pointing at the whole journey itself.
In #1, by using starting out, it points to the fact that the career had just begun - time reference, not necessarily to the length of time.
Again, #2 and #3 points to time.
#4, puts more feeling to the discussion as a whole, not that it had just begun.
#5, again, whole exercise experience here, also implying that things will change over time.
#6,#7, Journey and contrasting beginning to end with statements.
The last one: "Start out by accessing your list..." again is pointing to a specific point in time - when is being emphasised more than what you are doing.
Interchanging "off" for "out" on the above are all still valid, but changes (and only slightly) where the writer wanted the focus.
This is all personal conjecture here and I welcome other opinions. I really had to think hard about each reading to get a feel for the difference here, which should give you an idea of how subtle this topic is.
arduous has quite strong negative connotations: it is a word that you would use about a task, but not about a person's contribution to completing the task. It means difficult, needing a lot of effort and energy.
When the land rover broke down, we had an arduous trek to get back to base.
His duties are arduous and require a great deal of bending, squatting, reaching and lifting.
He had almost completed an arduous 16-hour shift when the accident happened.
A more appropriate word would be enthusiastic. Which in this context committing a lot of enthusiasm (energetic interest) to helping you.
Best Answer
When using the adage heart out you imply that the action is very intentional and the person doing the action is putting all their effort into it.
When using the adage *head off you imply that the action was done excessively, but not necessarily purposefully.
To laugh your head off makes sense. Genuine laughter is uncontrollable, and during certain moments it may come excessively.
To laugh your heart out does not really make much sense, unless the circumstances are that the person is attempting to laugh purposefully and is doing it with great effort.
Take the word sing for a reversal.
To sing your head off does not really makes sense, unless the context is an occasion where excessive singing is what your were trying to do.
To sing your heart out makes sense. Wanting to impress your audience, you would likely put great effort into any singing performance.