Can I activate an empowered Heimerdinger, Inventor to ready 2 gear if my Jayce, Defender of Tomorrow legend isn’t empowered? What about if my Heimerdinger isn’t empowered?

You can activate Heimerdinger’s ability if he is empowered, regardless of the empowered status of Jayce, Defender of Tomorrow.

There are two important details relating to dependent keywords:

1. A dependent keyword (the kind of ability that [Level][>] or [Empowered][>] are when they appear with the arrow) includes the whole of the ability, including the dependent ability. They’re all one object, so to speak.
2. A dependent keyword’s dependent ability still exists and can be referenced while it is inactive—that is, if its condition isn’t met. In this case, the condition is that the source of the ability is empowered.

Heimerdinger gains all activated abilities that have the [E] symbol in their cost, and he gains the entire ability, restrictions, costs, dependent keywords and all. In effect, Defender of Tomorrow’s ability that Heimerdinger is gaining is the whole line that says: [Empowered][>] [1], [E]: Ready 2 gear.

Heimerdinger can gain this ability even if Defender of Tomorrow isn’t empowered because of that second point.

The key rule is 727.1.b.1.: The Dependent Ability is present on the card for the sake of reference or evaluation until the Condition is met. This means that the ability isn’t active and its text isn’t being processed yet, but other effects can reference or evaluate its text before the condition is met. Heimerdinger, Inventor is exactly one such effect.