So this question I'm pretty sure has been answered before, but it's been all over the forum and I can't remember where I'd find a straight answer to this :P.
How do damage-over-time spells interact with weapon speed?
My understanding is that by default, a DoT will tick once per second. However, this rate of ticking is affected by weapon speed. So if, for example, Locust Swarm lasts 8 seconds, then with a fast weapon, Locust Swarm may even finish all eight ticks in 6 seconds or something, right?
Is there anything else I should know about how various weapons affect the way DoTs work?
Ok here is answer for you. Lets take the barbarian for example. The benefits of dualwield is that you get more spirit regeneration, and its better if you are planning a build that will be focused on killing bosses with stun on hit skills (like bash and frenzy runed versions) due to the faster attack speed. So, it is logical to assume that wielding a two hand weapon should have its own benefits, besides the higher physical dmg.
And that benefit is when using DoT spells, such is rend and earthquake.
And no, dot skills dont tick once per second, take rend for example from the youtube barb playthroughs. It is constant dmg over 3 sec, the only benefit that this skills could have from dual wield as opposed to a two hand weapon is the faster casting time. But that is irrelevant for this spells, definitely a 2 hander is the way to go if planing to use a lot of rend and earthquake or any other dot skill.
Also take notice that the speed at which DoTs deal damage and the rate at which the damage is displayed aren't the same thing. Damage is shown every 0,5 seconds, can't remember how it's rounded but anyhow it's rounded to full numbers. In reality a skill can deal for example 3,65 damage, and that would display a hit of '4'. But this is more of a non-DoT thing
For Earthquake and other DoTs with cooldown, the damage is based on your weapon speed too (= so your weapon dps, the damage of hit * APS), or slow weapons would be too good for those skills in comparison. For spammable dots this isn't a concern since faster weapon means the DoT in question can be spread to every monster around much faster than with a slow weapon, so they balance out.
More info about scaling of different DoTs in this thread.