For most end users, the answer is “snapshot 18w43b and above, releases 1.14 and above”.

In theory, there is nothing stopping you from running Fabric's mod loader on older versions of Minecraft, all the way down to c0.0.11a. However, mappings don't yet exist for those versions - as such, making mods is made… a tad more complicated.

As of now, no. There is some experimental functionality available in Loader's 0.5.0 development branch to run Fabric Loader on top of Paper specifically, but there are many caveats pertaining to Bukkit API's deficiencies. This will probably never be officially supported and should be treated more as a curiosity/special-case scenario.

With regards to modern Minecraft versions (1.13+):

• Fabric does not currently work on top of Forge on those versions.
• It is theoretically possible to create a way to do so - that is to say, there are no known major technical obstacles in accomplishing such.
• There were some experiments and discussions done in this regard (as of 30th June), but nothing usable by end users or mod developers.

With regards to older Minecraft versions:

• 1.6 ~ 1.12.2: While Fabric used to run on top of LaunchWrapper, this functionality is currently unmaintained and in need of rework.
• 1.3.1 ~ 1.5.2: Not currently planned.
• 1.2.5 and below: Functional! See the ModLoader section.

The development team does not consider Forge interop a high-priority goal, as our focus is on supporting the Fabric community and its developers and users.

Yes! However, to ensure proper support, you must enable compatibility mode in Fabric Loader, by adding the JVM flag -Dfabric.loader.useCompatibilityClassLoader=true. To some extent, this also works on Java 9+ - however, some mods, like RedPower 2, will break.

Not officially. However, unofficial mods exist to faciliate the launching of OptiFine for 1.14+ on Fabric. Your mileage may vary, however.