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.
Both sentences are correct but the underlying message is like you are asking them a confirmation from you if you are okay to start on that date, not a confirmation from the management. It will be redundant since you already have confirmed that you are willing to start on that date.
I think you should reconstruct your message to something like asking for a confirmation if the said date (June 10) is the final date of your first day so you can book your flight.
Best Answer
Neither of OP's alternatives are acceptable compressed forms of what would normally be referred to as date of starting work in the full form of this compound noun. The most common short form is...
Note that the base noun here is date - modified by another noun start, which is itself modified by the noun work. Most compound nouns involve only two nouns anyway, and if there are more than two they usually all modify the same base noun, but the only grammar involved here is that when a noun is being used "adjectivally" it follows the normal rule for adjectives - it comes immediately before the noun it modifies (even if that modified noun is then used adjectivally to modify another noun that follows).