As funny as it sounds, I was not able to find that answer on the internet. So before wasting all those precious level point in charisma I would like to understand something.

1. When opening, let's say, a "Weapon Shop", do you need to put the weapons for sale in the for the settler to sell those. Or the shop is self managed and you just collect the caps? I am pilling all those weapons for a while, wander if I will be able to place those in the booth for sale...

Mostly shop is self managed and you collect caps - based on tier of shop and population in settlement.

You can place some weapons in their inventory - legendary and unique weapons seem to take a really long while for them to get rid of. But it doesnt affect their money making ability. It is more to alow you to buy it back if sold accidentslly i imagine.

Also - all stores of same type in the same settlement share common caps and inventory.

Two general trade stores will provide double the happiness bonus - but cap supply snd inventory is shared. So happiness is per store, inventory/caps is per store type.

When talking about the Legendary weapons ... you said it takes a long time for the store to get rid of it.

So it DOES mean that you can place weapon in the inventory with the intention to sell those. If its the case, what is the price given back by the system? I guess it is random?

You buy weapons from the weapon store, you do not stock it.

I believe the income from stores is based on how many unassigned settlers are at the settlement.

More importantly, you convert the Purified Water from your Water Farm into Caps, Salvage Items, and Ammo at the stores. The Caps generated by the stores themselves is negligible.

How do you make money off of water purifiers?

Collect the water and sell it to a vender.

If the vendor is in your settlement you get to avoid the fast travel loading screen.

I think it's actually juts removed when enough time passes for the inventory to reset. I don't think it actually gets sold. I could be wrong though.

You sell purified water, HugeBullFrog.