Before asking any of these questions, you need to decide which device family you want to use because the answers to some of your questions differ based on that (only Arria 10, Stratix 10, Cyclone 10 GX, and Agilex supported in Pro; Arria 10 is supported in Standard as well).
1) Correct though it's no guarantee that a third-party IP listed there would support tethered mode.
2) Yes, IP Base suite licenses are included with Standard and Pro.
3) The e core is available in all editions.
4) Check the licensing center for the core(s) you're interested in: https://www.intel.com/content/www/us/en/programmable/support/support-resources/support-centers/licensing.html