Trading for quests is a thorny issue if you don't know exactly what you're doing.
For Belle Starr, you have to sell various types.
For 'Sell To Town Members', you need to actually sell to a member of your town AND when you sell, you need to put, 'Sell To Town Members Only' then - when your town-mate buys your items - you have to pick up the money to complete the transaction.
Likewise, when it says 'Buy From A Town Member", you need a town-member to put something on the market, saying, "Sell To Town Members Only", then when you buy the items, you need to pick up the items to complete the quest.
The same applies for "Buy From Alliance Members" or "Sell To Alliance Members" - they/you need to put "Buy From Alliance Members Only" or "Sell To Alliance Members Only".
The last part is the easiest.
To complete the Buy/Sell On The World Market, you just need to sell something that is listed as "Sell To Anyone" or buy from someone who has listed items as "Sell To Anyone."
Once again, you need to pick up the money/items to complete the quest.
For the Wandering Trader (or whatever he's called these days), you just need to buy from him, or click the Sell button and sell to him.
BE WARNED though. If you need to sell more than 1 item, do NOT sell MULTIPLES!!! They only count as
1 transaction. If you sell him 100 Trout for instance in 1 transaction, it COUNTS as 1 transaction...not 100.
I hope that helps everyone who has any queries about this and any other quests where you need to Trade.