There's a project now to include anbox to run Android apps http://news.softpedia.com/news/ubuntu-phones-will-soon-run-a...
I do use lineageOS myself and am generally very content with it but I am always a bit concerned when I see that the data usage of OS components (which are thrown together as one "app" in the settings) is more than a few MB. The system must do more than just checking for updates. Unfortunately, you can't prevent system components from accessing the internet if you don't intend rooting your phone.
Supported devices: https://wiki.lineageos.org/devices/
It turns out many apps from the Play Store don't actually have a hard dependency on Play Services. Example: Slack works, except for notifications and (IIRC) receiving calls, which is good enough if you don't need to be reachable all the time.
I run LineageOS v13 now (Galaxy Nexus), and before that Cyanogenmod back to v7 (maybe v6 not sure) and have never had Gapps installed, its been F-Droid all the way. I guess that means I can't appreciate the improved battery life or speed..