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.
Yes it will count for Ikora part III, since you are still doing the weekly heroic strike, there is no limitation if you have already done it this week.
Just remember you won't get any loot for it, it will only count for the mission.
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Best Answer
Status of your nightfall completion does not matter for that bounty. So you can do that bounty on your second run, or by just redoing first part of the strike or however you like. Just start nightfall and get to killing some majors, that's all.
As for the frequency of that bounty... Personally, I don't remember any week when we had two of those, but I could miss some. As for the future, Bungie controls the algorithm of bounty generation, and as far as I know they didn't disclose any information about it.