Hi ,
1) Gtm provides intelligent load balancing/DNS which for which it will reference the pool and give resolution accordingly .
GTM will listen for dns request on listener ip defined on the GTM .
And there sholud be connectivity between self ip of the GTM 's so that can communicate through iquery .
"How is traffic load balanced at the global level - is it just a case of users automatically connect to their nearest GTM?"
For this type of load balancing you can set Topology based load balancing .
what's the purpose of the GTM's having visibility of each other?
GTM need to communicate with each other if they are in sync group or meaning that they are sharing configuration . To keep config in sync , to know state of the other devices etc
if it is statndlone and are not depend on the other GTM's it is not necessary .
GTM will be load balancing according to the poad balancing method which you have set .