Yes. If you choose the free dogma action when you bring a card out of your forecast standard sharing rules apply.

In particular, the rules state that the player 'may' execute the Dogma after Melding a card from their Foreshadow pile.

If the player chooses not to execute the Dogma, is there still sharing?

Ok, get that.

What if you DON'T choose to do the free Dogma action when bringing a card out of your forecast pile? Do other players have the opportunity for Sharing?

I am assuming not, and that is how we played it last night, but would like the clarification.

After a meld action (only an action, not when you meld due to a dogma or echo effect), you may choose one card from your forecast of equal or lower value to the card you melded. You meld that card immediately, and then you may take a free dogma action, selecting that card to activate.

The free dogma action due to a promoted forecast card behaves exactly the same as a normal dogma action.