I will start answering to flush out the semantics described according to my sense of American English.
In "1. I have to go now, my Uber driver has arrived," the possessive is normally required and is a compact way of saying "the Uber driver that is coming to pick me up has arrived." If the context is absolutely clear, it is acceptable to say only "the Uber" driver, but it would still seem to call for a very specific discourse referent, implying the listener knows specifically that an Uber driver was called only on your behalf. Pragmatically, since you may not immediately recall what shared referent you have with your conversation partner about your departure, it is safer to say "my Uber driver." You could say this even if you have not said that you had called an Uber or if you had discussed using a different car service, or even if you had not mentioned that you were leaving at all. "My Uber" has much broader reference than "the Uber."
In "2. So, have you already learned your ABCs?," using the possessive asserts that learning ABCs is already part of the addressee's curriculum. If you say "the ABCs," you leave open whether or not the person will ever have to learn them. The former is what is usually said to a child to generate a feeling of shared experience, implying that everyone learns the(ir) ABCs as part of their normal development. Mentally, it is tied to other similar expressions like "Have you learned to tie your shoelaces?" or "Have you learned to open the lock on your locker?" or "Have you learned to brush your teeth?"
In "3. I now will put my eggs into the dry ingredients," the possessive is completely optional in this case, unlike in the other two sentences. You could also say: " 3.b I will now put the eggs into my dry ingredients." I think using the possessive helps to manage how the listener shifts mental focus. In the first sentence, the speaker implies a joint focus on managing the eggs as they go into the dry ingredients, whereas the second implies a continuing focus on the dry ingredients as eggs are mixed with them. Both project a certain immediacy and the shared experience of speaker and listener. If you use no possessive, you merely give an objective description of the recipe as is usually done in a cookbook, as opposed to a live performance where you can be more inclusive of the audience.