Looking over the diagrams in "The Egg of Time" and comparing them to what Vivec says about Kagrenac's Tools and the Heart of Lorkhan, I think it's possible to deduce how the Tribunal and Dagoth Ur became gods.

The first diagram in "The Egg of Time" depicts the figure of a well-dressed humanoid as well as the Heart of Lorkhan being struck by Sunder and Keening, which matches Vivec's description of the first two phases of the process of becoming a god: "The normal procedure for establishing connection with the Heart is a three-step process. The wearer of Wraithguard strikes the Heart with the hammer Sunder, causing the Heart to produce a pure tone. Then the wearer of the Wraithguard strikes the Heart with the blade Keening, shattering the pure tone into a prism of tone-shades. These tone-shades are then imprinted upon the substance of the wearer of Wraithguard, giving him an immortal and divine nature."

The second diagram in "The Egg of Time" depicts the same humanoid figure apparently being incinerated, and the Heart of Lorkhan being struck twice by Sunder and twice by Keening. "The Egg of Time" is supposedly a refutation of an argument that using the Heart of Lorkhan and Kagrenac's tools is dangerous, so I think that figure is not really being incinerated but rather turning into a god.

So my theory is that the third phase of using the Heart of Lorkhan to become a god, which Vivec refused to divulge, was to strike the Heart of Lorkhan again with Sunder and again with Keening. Either that, or the Dwemer killed themselves trying that.

Did the guys at Bethesda consider that a player might try to strike the Heart of Lorkhan with Sunder, Keening, Sunder again, and Keening again (or any combination of Sunder twice and Keening twice)? What happens if you do?

In an unmodded game, nothing. There is a mod out there called "The God Mod". I've never tried it but from what I understand, the author implemented something similar to what you are describing.