I can comment on Amazon Aurora:
Is the product a fork of PostgreSQL or a wrapper round the current version?
Aurora is a fork: they've re-written a significant chunk of the engine. Note that Amazon also offers RDS PostgreSQL, which is a managed version of the "regular" PostgreSQL engine. RDS PostgreSQL also offers a HA setup (no version upgrade without downtime, however). It works quite well.
Will the DB engine keep in lock-step with new PostgreSQL releases or might they diverge?
Amazon promises to keep it in lock-step. How soon they will release an upgrade to a major version remains to be seen.
When new versions of Amazon Aurora/Azure DB for PostgreSQL are released will our live instance get automatically updated or will we be able to choose a version?
Minor version upgrades are applied automatically. For major version upgrades, it's unclear at this time (there hasn't been one yet for Aurora PostgreSQL), but I think it's unlikely they will be applied automatically.