I have put this weapon in a level 1 adventure. Odds are the PC will not find it but you never know.
+3 SYLVAN WEAPON: This +3 weapon, when used outdoors in a woodland climate, inflicts an additional 1d6 of damage on a hit. If a target is struck to 0 hit points by a single strike, the wielder of the weapon gains an attack as a bonus action on another single target within melee range of the weapon’s wielder.
How would you reword this for 5E? Its from Castles and Crusades and I might require that it requires attunement by a Druid or Ranger otherwise it functions as a +1 sword. Druid melee damage sucks anyway so I am not to worried if one of them gets this and I will likely make the blade a short sword or scimitar (probably scimitar).
If the stats are from a game where magical bonuses go from +1 to +5, the first thing to establish is that the weapon's bonus is "medium".
Medium in 5E is +2, not +3.
Then, since the weapon has other abilities, it would be perfectly reasonable to tone down the plus bonus, since there are other effects. Unlike, say, D&D 3.0, there's only one main "layer" of magical pluses: a weapon is either magical or not; a weapon either cuts through monster magic resistance or they don't.
Even a +1 weapon that does a bonus d6 damage sometimes would be a highly desirable item that can easily be the basis for legends, stories and adventures.
One final piece of mechanical advice: don't make the free attack when you down a foe a bonus action. (Obviously, a "bonus attack" in Castles & Crusades might mean something entirely different than the highly definied "bonus action" in 5E; I'm going merely by how similar the terms sound)
This shafts two weapon fighters. It shafts users with the GWM feat. It shafts rangers who need that bonus action to shift their Hunter's Mark.
I suggest you make it a free additional attack, so everybody can get the same bonus out of it.
Instead of rangers, one of your intended user, not being able to use its defining feature, since they're among the most likely character to use TWF and/or Hunter's Mark.
Also, making it a free immediate attack is in the spirit of the original's "within melee range of the weapon’s wielder". If you make it use the character's bonus action, nothing stops him from moving into range of a new foe first. I mean, you could state this can't happen, but making an exception to this general rule is not worth it in my opinion. If you instead say "you get a free attack to use immediately without moving against a target within reach" that's much less of an anomaly in the 5E framework.