re: Quests. I think in the description for Breon's Letter (or the one for lvl 7 champions)- the description implies that you need to complete this research to start the Epic Quests that fulfills the game's victory conditions.
You no longer need to reseach the techs to unlock the quests since this beta version. The descriptions of many techs and buildings have just not yet been updated to reflect the changes made to them, because their effects are not yet final.
re: weapons; the Ice spear and regular spear does the same normal damage with the exception that the Ice spear does add'l Cold Damage. So the regular spear cannot be the 'best' possible weapon.
The game regards the weapon/armor which is the farthest to the right in the Unit Design Screen as the 'best'. It doesn't matter, if the item really is the best one available.
I researched Advanced Agriculture (or whatever its called exactly) and when research was complete it did not register that I had researched it. I selected it again, but game ended before it finished.
This is not a bug. The Advanced Techs can be perpetually researched and their effects stack.