+atk/-spd regular Lyn?

Yes she works fine with a spd bane.
Not really, her refined weapon has brash assault 4 effect built into it. So as long as the enemy can counter she will always double them regardless of their speed (unless they have swordbreaker).

Lyn will work fine with a speed bane, it's not ideal, but she'll still have 34 speed, which is workable for avoiding doubles. As long as you refine her weapon her player phase will still be very strong, although obviously she will be a bit more prone to be more doubles in the enemy phase against super high speed units (like mia, ayra etc)