If I use Challenge while my unit is in base and I conquer an open battlefield after that, do I heal?

No, you do not heal in this scenario.

According to RiftJudge FAQ #2241 and #7796, there is a critical distinction between a combat showdown and an open showdown:

• Combat Showdown: Occurs when a unit moves to an opponent-controlled battlefield. This triggers combat, which includes a "Combat Cleanup" step where all units heal.
• Open Showdown: Occurs when a unit moves to an empty (uncontrolled) battlefield. This is a showdown, but it is not combat. Because it is not combat, there is no combat cleanup, and therefore no healing occurs.

Since you are conquering an open battlefield, you are engaging in a non-combat showdown. As stated in RiftJudge FAQ #1209, "Non-combat showdowns do not have combat cleanup," and without that cleanup, your units do not heal.

Additionally, regarding your use of Challenge:

• Challenge is not combat; it is simply an effect that deals damage.
• As noted in RiftJudge FAQ #7008, units do not heal after using Challenge. They remain damaged until they heal after combat or at the end of the turn.

In summary: Moving to an empty battlefield does not trigger the combat cleanup required to heal your units.