How fast do you kill the boss?

As far as I am aware, most skills operate on the damage type you've converted to, because most of them trigger off the damage dealt. There's even a couple of small effects which might seem like they apply to the original, but don't (for instance, the heat passive tree keystone skill applies *only* to heat damage, not physical damage dealt using energy weapons). The physical passive tree node that applies vulnerability on physical crit does not operate when using the convert-to-warp doctrine.

I've not tested the compound case, so if you have both components, I'll defer to you, but I would wager that---again---it is probably not like PoE where there is a unified mechanism. It is more likely that one simply subsumes the other, and even if it does not, since most things are keyed off the damage dealt, the end behavior would probably end up being similar from the outside. Likewise, I would suspect that the armor-breaking doctrine would cease to apply if you converted to something else.

That all said, I've only personally tried some of the possible cases, and I am not privy to the implementation, so if you do end up doing more experiments on the question, please let us know! I'd love to know what you experience.