We generally enable compatibility to upgrade from the latest in any given version to the next. However, you have a really really old release you inherited!
Question 1
Cannot go directly from 2.1.4 to 2.3.x-- you'll need to go through the latest in the 2.1 series. No need to go to 2.2.
Question 3:
Looks like you can go from:
2.1.4 -> 2.1.latest (looks like 2.1.8)
2.1.8 -> 2.3.12 (doc: 23.4. Upgrading - - The Neo4j Manual v2.3.12)
2.3.12 -> 3.3.9 (doc: 5.1. Upgrade planning - Chapter 5. Upgrade)
3.3.9 -> 3.5.5 (doc: Upgrade checklist for 4.x patch release - Upgrade and Migration Guide)
Question 2: Backup and restore is available on enterprise (and backups are a good way to do upgrades there), but not on community. The way you mentioned is the best way to do it on community.
Cheers,
-Ryan