Does the hot stack up with a previous eternal flame hot? ie does it have diminishing returns to use it on the same person twice?
Does it snapshot stats when it is casted? Does it snapshot buffs like Divine favor or Wings or does it update dynamically? Does it scale with crit and haste? If so, is it like the lock doomguard and only scales with rating not actual %'s?
Same as Lights hammer. Also since execution sentence is a "spell" and Light's Hammer acts as a "pet" does that mean they snapshot stats/scale differently?
The only answer I personally can give is that snapshots were all taken out as far as I know. I might be wrong, and our spell may be the same, but they added the fact that the spell changes as your stats change, I know for a fact that pets are included. I know from a dmg perspective that ES chances as buffs wear off or come up. Also I don't believe EF stacks, nor does it suffer from diminishing returns.
Using Eternal Flame on someone who already has the Eternal Flame heal will refresh the HoT. (Note that, because of the below, if you have less potent stats, you'll actually replace that HoT with a weaker one, albeit with a full duration)
Both Light's Hammer and Execution Sentence do snapshot your stats when you use them. The "change" they made from before was that whenever a buff or debuff is refreshed, it used to keep the original stats used to apply the condition (spell power, haste, crit, mastery, and so on) even if you no longer had those stats - thus, a Warlock could burn all of his cooldowns to apply all of his DoTs at full power, and then as long as he just kept refreshing them, he would "roll" those huge DoTs throughout the entire fight. Now, it's been changed so that any DoT that's refreshed or re-applied actually redoes the stats when you refresh the DoT. This means that long-lasting or long-cooldown DoTs can still somewhat benefit from big stat boosts (if you refresh them just before the stat boost wears off) but it can't just roll throughout an entire fight. You want to use Execution Sentence when all your buffs are up; Light's Hammer has a short enough cooldown that you don't have to specifically time it, but it's still good to throw it down before your trinkets, potions, or engi gloves stop being in effect.
Using Eternal Flame on someone who already has the HoT isn't the most ideal usage of Eternal Flame (as the HoT portion is partially wasted), but since it's replacing Word of Glory, it's never bad to use it on someone if you have to - it heals for more than Word of Glory directly, plus you'll still get some extra use out of the HoT.
I know for a fact that Light's Hammer can be modified by Avenging Wrath after it's been casted. I've tested this before, due to the fact that for the first ~2 seconds of it being down it does 0 healing.