Is Toxikon a dps gain?

Toxikon is DPS-neutral on one target at level 90 and a DPS gain on 2+ targets if you already have a Toxikon charge. Using Eukrasia Diagnosis and then Toxikon is a loss on multiple targets compared to Dyskrasia.

What is the purpose of Pepsis, will I be using it a lot?

No, Pepsis is a utility button but not something you will be using as part of your healing rotation. It’s a good way to clear shields if an incoming AoE needs it and your shields may fall off right before it happens.

Pepsis also has a niche use where you can use it to double dip by abusing snapshots. That is, use Pepsis between the damage snapshotting and applying. The damage has already snapshotted, so consuming the shield at this point doesn’t reduce its effectiveness, but you will also get the Pepsis heal on top of it.

Can I double weave after my DoT?

No, Eukrasia has a 1s recast, while the following spell has 1.5s recast time. You can only single weave after the second spell.

Do shields stack?

The image below shows conflicting shields and which shields win out, anything not listed will stack. (Thanks to Meru for the image.)

How do I fit Phlegma into 2-minute buffs?

Now that the cooldown has been reduced from 45 seconds to 40 seconds in patch 6.3, fitting Phlegma into buffs is easy. Every two minutes, you’ll receive three charges total. This allows you to put two of them into buffs, and use the third outside of buffs. Assuming your party is doing a normal opener and bursting every two minutes thereafter, you would use your first two charges as soon as raid buffs go up. The third charge is free to use anywhere from 0:40 to 1:20, and then you’ll have two charges available at the next two minute window.