A bit tricky because there are so many phrasal verbs with “get.” I’ll try to clear up some of the confusion.
“Get to [verb]” (for most verbs in general) - typically means “to have the opportunity to [do or perform said verb]”. For example, “I won backstage tickets to this concert. I can’t believe we get to talk to our favorite artist!”
“Got to talking” - colloquial/informal for “started talking,” most often used in situations such as your example (“She was the most beautiful woman I have ever seen. I made up some pretext for asking her about the time, and she seemed interested in continuing the conversation, so somehow we got to talking, and now here we are celebrating our first anniversary!”)
“Get to work”/“get working” - phrasal verbs used to say “start working,” most often in exactly the type of situation you described above (“we have a lot to do, so let’s get to work!”). “Get to work” = a little more common. If you say “get working” people will understand what you mean but it is not used as often.
As you can see, “get to work” is a phrasal verb that does not have the typical meaning of “have the opportunity to do”. However it MAY also be used in the first sense, as in “they changed my schedule, so now I get to work [have the opportunity to work] on Thursdays and Tuesdays instead of just Tuesdays.”
Also, remember that “get to” can also mean “arrive to/at.” So “get to work” can also mean “arrive at work,” for example “I get to work at 6am every day.”
Finally, the clauses in examples 1a and 1b I would connect differently; maybe something like “she was the most beautiful woman I have ever seen. We got to talking, and... [continue the train of thought].” Separating the two clauses by a comma and “then” doesn’t quite make sense to me, but maybe it would make sense with the context of the sentence.
Let me know if you have any questions! “Get” is used in so many different phrases, it takes a lot of listening/reading to really *get* it. :)