There's no way to determine if an item is safe to sell without looking it up online. Even items which have descriptions which make it sound like they could be sold might still have a use. In fact, the Silver Egg is a perfect example of this. The item description says it's "of no use to a hunter", but it's used to craft Fate Jewel 1.
Kiranico is a fantastic resource for any hunter, and contains searchable information on all of the items in the game. It's your best bet to determine if a specific item is safe to sell or not. The sources tab shows you all the places you can get that item from, and the usage tab shows you the various uses of that item. If the item shows up under the uses tab, then it isn't safe to sell. The uses tab doesn't list all possible uses though, strangely enough. So you'll want to check the Veggie Elder, Hunting Fleet, and Villager Request pages as well. Finally, you'll want to double check this page on farming, to see if an item is also used for upgrading the farm or acts as a fertilizer. If the item doesn't show up on any of those pages, then it's safe to sell.
Here are some links to read more about the Veggie Elder, Hunting Fleet, and Villager Requests, just in case you're not familiar with what they are.
I've compiled a short list of some of the items you can safely sell, but this is not a complete list:
With that said, if your main concern is making money, then these safe to sell items aren't how you'll make most of your cash. The best way to make money is by selling monster carves from large wyverns (Great Jaggi, Rathian, Barroth, et ceter). These items are used for crafting weapons and materials, but unless you plan on crafting every weapon and item in the game then you're not going to need every carve from every monster. Kiranico is a good resource for determining which armors/weapon you want, and by extensions which monster carves you don't need.
Farming is another good way to make money, although by the time you hit G-Rank it'll be completely outclassed by monster carves. The Monster Hunter wiki contains a good article on farming which I'd suggest reading. The important thing to take away from it is that farm yield is increased by using fertilizers. Using fertilizers to increase yield is very important if you want to make a good profit from farming. Nulberries are my preferred item for farming. Just check your farm every few hunts, replant a few Nulberries, and then sell the rest.
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.
Best Answer
No. You cannot capture a monster by inflicting it with the Sleep effect. Inflicting Sleep is not the same as Tranquilizing a monster, as they are two different things.
You have to use Tranq bombs or Tranq shots to capture a monster, assuming it's weakened appropriately.
See posts like the ones in threads like this: