Yes.
Your assumption is correct … it’s Android. Just install F-Droid and your usual Apps and you’re good to go … as long as your Apps work on newer Android versions.
Yes.
Depends on your priorities.
The current LineageOS 16.0 is Android 9.0.
The upcoming LineageOS 17.0 is Android 10.0.
While I always insist that a version number alone is not an improvement in itself, LineageOS certainly feels more up-to-date. While it would be debatable whether every change in a new Android version is for the better, LineageOS is helped in this regard by Fairphone’s (understandable) troubles with Android 7. To improve on that seems a low hurdle to clear if you disregard Google’s Android certification, like LineageOS does.
Regarding updates …
Fairphone’s updates don’t come that often anymore, but they are beta-tested before release.
LineageOS updates come more often, but the releases are automatic nightly builds you install at your own risk. LineageOS aims for stuff to not break, naturally, but automatic builds are automatic builds. If you want to feel secure, perhaps somebody takes the plunge and test drives a build a little (Example with LineageOS for microG, which is a LineageOS fork with microG robustly built-in, for those who want to have that).
As can be? I don’t know. Every Android is based on Google’s work, and there are always still bits and pieces here and there which can be discussed.
Let’s say: If you were content with Fairphone Open OS in this regard, then there’s no harm in checking out LineageOS.