a) If I complete quests incorrectly, does that stop me from completing a chain, and will I have to start over?
You can't technically complete a quest incorrectly as that implies you manged to complete it in a way that wasn't designed. You can get what I prefer to call "non-optimal quest endings", these will effect length of quests, what you get told settings wise, rewards.. etc
Repeatable quests can be discarded but you are still subject to the timer which blocks the time between which quests can be started again and this varies for most quest. Repeatable quests, we have also been editing to make much more variable. Take Remant's gauntlets which can now be lost in many different places each time you talk to him about it, or Harnquist is hungry which asks for different things (also based on cooking and baking skill level) where the more complex things he asks for give much better rewards. The only thing to be careful about is if you discard a quest the next quest in the chain will not be accessible until you complete it again as the quest has to be a "completed" state.
Contrary to what Ligh said there are -some- but not many quests where specific endings are necessary to progress down a quest chain. But the vast majority have nothing like this. We have some quests being made where the ending you chose will change the next quest but this will just be for small 1-2-3 quest branches.
b) How can I restart a quest now? Or can I not?
If you chose a "non-optimal quest ending" it will not be reset for you. There would be no point in having endings like this if you could just get them reset. The only case where quests will be reset are when there are bugs in the quest preventing you from continuing. Due to bad design, or a change in quest design which isn't compatible with people who started the old version. The last time these bugs appeared was when the questnotes were added a few people are still finding they are stuck like this. If you lose a quest item you can just ask for a new one and a GM can help you out.