These are good questions, thanks!
1: Math was obvious; it's always been my favorite subject and I felt like learning more math was the best way to (a) leave my career options open and (b) improve my general critical thinking skills. Going into college, I was undecided between economics and CS, but I decided to major in economics after taking a few great classes and because even as a CS major, I would have been able to take only a limited number of CS classes because the Swarthmore CS department faces significant enrollment pressures. I also realized I was much more interested in CS theory than in most other fields of CS, and most of the theoretical CS classes were underenrolled and were thus not too hard to take as a non-major. (I do wish I had the chance to take a class on programming languages--I've heard from some friends that PL theory is really cool.) Overall, it was a marginal decision and I probably would have triple majored if Swarthmore allowed it.
I really enjoyed the college experience. My math/CP background made it a lot easier to succeed academically, not so much because I had prior exposure to some advanced topics but mostly because (a) I had a very strong grasp on the basics, meaning that I didn't get tripped up by e.g. basic algebra and was able to instead focus on high-level problem solving, and (b) psychologically, I was already used to spending time thinking about hard problems and had practiced dealing with problems I didn't immediately know how to solve. I especially enjoyed taking math seminars (at the introductory graduate level) in my last two years. Outside of school, I made some close friends and was able to spend a lot of time having fun with them, especially after campus reopened post-pandemic.
2: I essentially treat competitive programming as a hobby, and e.g. I generally don't try to force myself to train unless I feel like it. It's not involved in my career directly, but I think my competitive programming awards were a significant part of why I was hired for my current job, and they'd probably help if I eventually decide e.g. to go into software engineering or work for a quant firm rather than pursuing a Ph.D. I've also coached at various points to earn some extra money on the side.
3: Starting in 2019 when I began to take CF more seriously, I was mostly focused on the next step at any given time, i.e. I didn't have LGM as an ultimate goal for the whole time. In early 2019, I realized I could probably become red and eventually made GM that summer. At that point, I pretty much immediately started thinking about IGM, and while I had a couple of significant setbacks I crossed 2600 the following summer. LGM took a bit longer, and I wasn't sure I'd make it that far until I crossed 2800 by earning +183 from a 6th-place finish in a Div. 1 round. At that point, I became pretty confident that I'd eventually make LGM and focused my training on crossing the 3000 mark.
I enjoyed the process of training (largely because I didn't force myself to do it when I didn't want to), and I don't regret putting in the time to become LGM at all. (In practice, I think my CF practice mostly crowded out gaming/other unproductive hobbies rather than cutting into my actual productive time, so I think the opportunity cost for me was not too large.) I especially enjoyed CP because I was able to practice mostly by doing actual contests--there were times when I would average four live contests or so a week between CF, AtCoder, CodeChef, DMOJ, etc. (I'm very competitive, so the feeling of winning contests/improving my rating was a big part of my motivation to improve at competitive programming.)
4: Yes, I was pretty thrilled. (There's an audio file, which I will not be releasing, of me screaming incoherently upon realizing that my 1:59:56 submission to D, which I hadn't run on samples, got AC.) At the time, I was expecting to make LGM eventually and I knew that each successive contest gave me a very strong chance at promotion (I was at 2973 in the round before I promoted), but once I actually pulled it off, it felt pretty amazing to finally achieve something that I had in some way or another been working towards for half my life (counting the time I spent doing math contests).
I did tell my family; I'm lucky to have very supportive parents and they stay pretty up-to-date on my programming contest activities. (As a particularly extreme example, my dad stayed up from 11 PM to 4 AM to watch the Dhaka WF live.)
5: The main way working makes training harder for me is that I can't compete in weekday rounds on days when I'm in the office. Otherwise, I essentially work a 40-hour week with flexible timing, meaning that it's not too hard to fit contests into my schedule. Recently, I've typically tried to think about a couple of problems during the week and then on the weekends, I compete in a live contest one day and do an ICPC VC with my teammates on the other day.
6: Back in early 2015, I spent a lot of time looking for a new Minecraft handle (a very difficult task because usernames were unique and most of the good ones were taken). I was learning about geothermal energy at the time and thought it was really cool, and since it was available I used it as my name. I've used it ever since, partly due to inertia but also because I think it works well as a username (I think it sounds pretty good, it's available on lots of platforms, it abbreviates nicely to Geo for when I'm talking to friends, etc).