1. Duplicates. I need the same unique card but the core set only comes with one of all the unique characters. Do I need to buy another whole set just to be able to duplicate cards?
Yes. Most players that enjoy the game end up with 2-3 Core Sets to allow for duplicates.
2. Milk of the Poppy. What's the point of this card? It just makes your characters weaker so why would anyone ever play it?
This card is best played on your opponent's characters to neutralize strong text (e.g. giving it to Robert Baratheon to remove his Renown, Intimidate, and strength pumping text).
3. Renown. If a renown character that has power on them dies, what happens to that power? Does the player get it on their faction card or is that power lost?
The power is lost. It pays to protect your Renown characters.
4. Deck-building. I don't see any way that I can build a deck of minimum 60 cards without taking cards from other faction decks. Is that allowed or again do I need to buy another whole set/expansion packs to get enough cards?
A legal 60 card deck cannot be built with a single core set - a single core is meant to introduce the game using the recommended small decks. Full deck building requires either multiple core sets or expansions. The Banner agendas (found on the reverse of the faction cards) allow faction mixing, but only allow non-loyal cards of the Banner faction.