Eclipse: Rise of the Ancients

Set it beside the Supply and ignore it when counting the regular Technologies to draw. You can buy a Rare Technology with a Research action instead of a regular Technology, but not as an Ancient Technology Discovery (p. 4).

Yes. All hexes with Warp Portals are considered adjacent and connected by a full Wormhole, so you can move, Influence, and form Diplomatic Relations through them (p. 8).

Allies cannot attack or pin each other, and may move into and through each other’s hexes. Their ships are combined when checking whether opponents pin them (p. 10).

No. You must destroy the Ancient Cruiser before placing your Influence Disc. Destroying it also lets you draw two Reputation Tiles, and it is worth 1 Victory Point at game end (p. 6).

At the end of the Cleanup Phase, roll once for each Hive; if the indicated side has an explored hex, move one Ancient Ship there. Hive Ancients can move through a half-Wormhole and are always defenders in combat (p. 7).

Yes—once per Ship activation. If you activate that Ship more than once during the same Move action, it can use the Jump Drive during each activation; it may also be the Ship’s only Drive (p. 8).