1. You can export NPC Memories and read through what they remember about any jobs, or quests they gave you. If you don't see any you do remember, then they forgot and you can consider any like that concluded. If you and any NPCs don't remember a quest, then it's not really worth worrying about it.
2. You can leave in the middle of any conversation, save the game and click back on the NPC to resume your conversation where you left it. This might not work if you leave while an NPC is talking to another NPC and you haven't interrupted.
3. I believe the pop-up dialogue box for renting a room is from the system and not Gareth. Though Gareth can change the price it shows if you haggle him down before it appears. Its not suppose to give money, but you can ask any NPC for money and they might give some to you. It might only be in RP sometimes.
4. Not at the moment, no. The best you can probably do is try adding to your character's description that they wear a crest that designates they're from (wherever) relative to Silverpine. Alternatively, constantly editing their memories so they always know is another suggestion, but it's a hassle to continuously upkeep and the more NPCs you want to do this to make it many times more annoying.
Your Skyrim comment reminded me of a video where the player was speaking to the NPCs with a mic and the AI was speaking back through the game as the NPCs in their voices with all of their background lore and world lore to roleplay them believable. It was impressive.