The changes to W only really matters for AP Trist, because, since it still has no AD scaling, AD Trist still shouldn't be jumping on enemies most of the time (its damage amp is only to itself anyway). I believe rank 1 does more damage base (or at least, not LESS), so it's still good in lane phase.
Her E is basically her old one, except it deals its damage at the end of the duration instead of over the whole thing, and it's damage is dependent on Tristana autoattacking, which is what she wants to do anyway. On the plus side, it has AD scaling now. On the minus side, removal of healing reduction makes her slightly worse in a duel against another ADC with lifesteal or during lane phase against a healing support.
Numbers-wise, her E gets reduced damage starting at rank 3 at 4 stacks, but that doesn't take the new scaling into account, or that it is now AOE and can affect turrets.
If you don't stack E at all, it does less damage at all ranks, but the fire-and-forget of her old E was a little too non-interactive.
On the topic of her attackspeed, I haven't played her enough to have an opinion yet, but the duration reduction on Rapid Fire combined with the nerfed base looks like it might hurt a bit.