1: I'm having trouble getting the appropriate quest-chain to get Pangu's Axe. Do I need a certain Visa level, or befriend specific Sims in China, or visit all of the locations first? Or maybe even see one of the unbreakable boulders before I know to get it?

2: One article said you need to win 50 ranked matches, and it can only be done in China. Is there anyway to work on the Martial Arts tournaments on the home lot? Or maybe even a mod? It just seems like a hassle everytime to work on it to go to China.

3: Do Elders always get the lame adventures? I kept cancelling and rerolling the adventures for a while, and it alternated between finding 10+ pieces of copper or photographing something in a tomb. I guess it's to prevent them from starting an important quest-chain before passing away?

Don't have answer for all of them, but

1. I think you need Confounded Boulder quest. Not sure on what visa level it will pop up, but higher the visa the better becouse the quest is long and three days isin't enough. The axe is located within the Temple of Heaven.

(1) Ever since Late Night the Adventure Board has been broken and it doesn't offer all adventures reliably anymore. Some of them will never appear and some may or may not appear randomly. If you think you should get one but you don't, no matter how often you try, you can test the Adventure Board for broken opportunities: the Adventure Board will always have 5 adventures ready at the beginning of a one-day period (starting at midnight IIRC), so if you queue up 5 "Check for Adventures" and your Sim simply either drops one or more of them (meaning it doesn't trigger the window pop-up where you can either decline or accept the adventure) or the first one reappears before you have cycled through all 5, then the Adventure Board is broken again.
Keep in mind though that some quest have requirements that have to be met before they can appear in the pool of available quests. The Confounded Boulder requires, I think, visa level 2. Whether there are other requirements in addition, I don't know. Personally I use MasterController to trigger opportunities. Under Basic you can trigger any opportunity for which your Sim meets all requirements, even if the Adventure Board doesn't work anymore.

(2) I think that NRaas Careers unlocks martial art tournaments in homeworlds.

(3) It's been ages since I've sent elders on vacation but it may well be that your issue is connected to issue (1), i.e. the broken Adventure Board.