Level downscaling will downlevel your character to the maximum allowable content level for that area.
So this will affect all your primary stats, Toughness, Vitality, Power and Precision. When you open your hero panel you'll see your main stats on the right in white, and next to them on the left you'll see your 'effective' stats in green. These are the numbers that you're going to be using. As these numbers are changed this will effect your damage, hp, crit chance, etc.
The thing with you feeling more powerful is that you still have the higher level gear. So for example, if you are lv30 you might have an amulet with +11 power. You go to a lv10 area, and fight with players who at lv10 can't even equip an amulet. Now I'm assuming, but let's say that the downscaling is a %, then your overall stats will be higher, meaning you'll be stronger at lower level because of your better gear.
You are by no means at all invincible, but you'll certainly fare a little better due to having more or better gear. The real killer is that as a higher level you will have traits and skills available which lower levels will not. You can also use your Elite skill still, if you've unlocked it. So there are some advantages to revisiting the lower level areas.
I think most people's experiences of this will be doing each of the race's starter areas once they've finished their own.
You can read more on the wiki, http://wiki.guildwars2.com/wiki/Dynamic_level_adjustment
Character level adjustments have a limit, however. The
level-adjustment system won’t let a level 1 character go head-to-head
against a level 80 character with much chance of victory—a level 80
character is still going to have more bonuses on their gear and access
to a full complement of utility and elite skills that players just
don’t have at lower levels. This is not to say that the high-level
character will one-shot the level 1 player, but they will have a clear
advantage.
http://www.arena.net/blog/mike-ferguson-on-guild-wars-2-world-vs-world
Edit
Since Jan 28th 2013, the downlevelling has changed. https://www.guildwars2.com/en-gb/news/dynamic-leveling-adjustment/
The official Guild Wars 2 Wiki has an article on damage (calculation).
Edit: I'll try to sum it up for you, as damage coming through depends on several factors.
The damage you deal can be raised if you score a critical hit: 1% crit dmg per trait point.
Condition damage can also add to the total. +10 cond dmg per trait point.
Effective damage you do is also dependent on the level difference. Then there are the mechanics of sidekicking and scaling, the Protection boon, positioning, projectile distance, ... that are all part of the complete formula.
Update:
The wiki article has since been updated, and now contains an easy to understand formula to calculate direct damage.
It is
Damage done = (weapon damage) * Power * (skill-specific coefficient) / (target's Armor)
Armor only mitigates direct damage, so condition damage and falling damage is not mitigated by armor.
Best Answer
On-kill sigils have stacks of up to 25 (you'll see a buff that stacks up to 25) which reset when you die. You should get hp when you kill an enemy, as well, as there is no reason for these two particular sigils to not work together as they don't share the same internal cooldown (ICD).