What are the requirements to unlock certain G3* quests

Excavated weapons are weapons found in Everwood expeditions. These weapons start off with another weapon as a base, but come with randomized stats.

You could conceivably find an Excavated weapon with much higher stats (or lower) than its base. They also have some slight visual differences, but it's easy to see which weapon they take as their base.

Since you're talking about Bows here, let's make this Bow-specific.

Yes, all damage (raw, elemental, status) is affected by the Bow's "sweet spot" range. Being in the perfect range will increase all inflicted damage by 50%. Hitzones do matter, so you'll want to target the same hitzones you normally do rather than fire willy-nilly. And yes, number of hits does matter for apply status. While the Monster Hunter damage formula is certainly convoluted and arcane, RNG does not play a factor outside of determining whether you made a critical hit with Affinity. Speaking of CritStatus works similarly to Affinity in that it gives a percent chance of dealing increased damage.

There really aren't any tips for inflicting status outside of what you would normally do. Perhaps if you dedicated to statusing, bring some extra combo materials. Other than that, just do what you normally. Note that if you get lucky with crits and hitzones, inflicting the same status twice is usually where diminishing returns kicks in, it'll be much harder to get that third status inflicted.