Here is what my experience on the topic is: During writing my own thesis I was focused quite narrowly on the topic and everything which would help me prove the results I needed. That already meant learning a lot of new (to me) and exciting (to me) mathematics. 
After completing the thesis my experience has been that my mathematical world gradually expanded. It often seems to happen to me that I stumble into talks and on results which at first sight seem unrelated to what I do and then reveal themselves to be connected to my own work (or give rise to new and interesting projects).
So to answer your questions:

1. Depends on what you want to achieve. Certainly you do not want to be completely ignorant at what is going on in mathematics or in neighboring topics. Having said this, it is very easy to get lost in the process of reading new fascinating math and loosing track of ones own projects.
2. I think it is good, especially at the beginning, to focus on some topic but keep an open mind. At least for me, working on multiple problems came at a later stage (and it might be argued whether this is keeping one from doing enough works on the project). However, this might be different from person to person.
3. Personally, I quite enjoy connecting different areas of mathematics. It seems to me that I am quite often discover new areas and am learning new things all the time (though that might also be different for other people). 

Finally, let me tell you that seniority has the benefit of experience in playing the mathematical game. On one hand it should come at no surprise that people who have been working years and decades in academia have a rich knowledge base on which they can draw to ask questions. Moreover, as one of my academic teachers liked to point out: A little knowledge about a field is more then sufficient to ask an intelligent question. While the senior people might not have detailed knowledge to work on a problem in a neighboring field, they certainly almost always know enough problems and keywords to ask something after a talk.

I can not say how typical this experience is, but what I suspect