To directly answer your question instead of quoting long winded and confusing reddit posts:
No, upping your light level will not make you more effective in terms of damage on content unless you were previously underlevelled for it, assuming the same weaponry. Your weapon reaches its maximum damage the moment you hit the same level as the enemy upon which you are firing (unless you're using severely underlevelled weapons, see below), and it will stay that way against all enemies you are of equal or higher level to. The only way to increase your DPS is by upping your weapon's attack or getting higher attack weaponry (or I guess unlocking certain armor mods, but that's besides the point).
Sidenote: This does NOT apply to abilities, just weaponry. Your abilities increase in damage as your light amount goes up.
You will, however, benefit from the better abilities that are on those pieces of equipment and the increased armor rating, which will allow you to survive better. You will also presumably have faster recharging abilities due to the higher level of stats on better gear.
The only way level applies to damage, then, is down scaling due to being underlevelled. If your level is below the level of the enemy you are fighting, you take sharp damage penalties, at four levels below you are not damaging the enemy at all (its level will show ??). There's also a damage penalty if your weapon's "internal level" is much higher than the enemy you are fighting, but that's rarely very useful information. It does, however, mean that you won't suddenly do worse against low level content just because YOU levelled up, if you're still using the same gun.
Best Answer
Bungie did not give much information about Three of Coins, besides a paragraph in the Bungie Weekly Update on 9/17/15:
So as we know it, this is how the buff works:
1) You must use a Three of Coins (3oC) to gain the buff.
2) Killing an Ultra (most strike bosses, most raid bosses, Taken champions in patrol, the Ultra Knight on the Dreadnought) removes the buff. On the kill you either get an exotic engram drop, or you don't.
3a) If you got an exotic engram, your increased exotic engram drop rate from 3oC is reset.
3b) If you did not get an exotic engram drop, the next time you pop a 3oC you have a better chance at getting an exotic.
Notes:
As of the recent hotfix, 3oC can't be used as effectively to farm. There is some hidden timer between popping them that affects their effectiveness as of the 9/24/15 Hotfix: "Three of Coins now provides a smaller increase in Exotic Engram drop chance when rapidly killing Ultras"
It is generally observed that the exotic engram drop rate to 3oC use is approximately 1:5, but YMMV. This is purely speculation based on accounts and sampling from friends, the Destiny subreddit, and the Bungie forums.
There are no known statistics, including the "cooldown time" alluded to in the first note, what percentage each coin raises your chance by, or anything like that. Bungie tends to be cryptic about these sorts of things.
It is also possible for most strike bosses to drop an exotic engram without a 3oC. This can be observed in cases of people getting two exotic drops from one boss. This hints, in my opinion, that the 3oC buff is a separate roll that occurs when you take down an ultra.