1. Even after completing the quest, the inventory still displays the first two gifts:

2. I got two identical jewels from the bandit camp at the sea. I suppose an exception should be added, so that the game generates only one copy of Ayda's necklace

Not a bug but a suggestion: it would be cool if the necklace were enchanted and Ayda (and only her) could equip it. Like Ayneris' rapier.

jup, aydas neclace need to have tag unique..since is not much point of  having them more , since they are usless, other that quest :)-

and i discovered bug with him, if it happen to have him as loot (funny my first copy was from wolves and not humans, another bug imo too) and you just clock on loot icon, quest will not progres, you actualy need to click on icon of that amulet to advance in quest.

and similiar proble is with that book/brandy - i acidentaly buyed more than just one - and now they are usless in my item..i will just dump them :). they should deserve unique tag too.

and sure, yeah i will love if ayda neclace will be (later?) usable by her. 

and it is similiar with ayneris rapier - will love if it will be usable only by her and not by anyone.

IF NOTHING HAPPENS WHEN YOU TALK TO AYDA EVEN THOUGH YOU HAVE HE NECKLACE(s) IN YOUR INVENTORY:

just as evilus suggested, instead of clicking the "Take all" arrow in after-combat loot selection, CLICK ON THE ITEM ITSELF when it is in the bandit's after-combat inventory!

a prompt should pop up indicating that you have pocketed the necklace. then you can go talk to ayda and she'll be happy.