These are complicated questions, as is anything to do with Summoner. For much more detailed answers and reasoning I definitely recommend The Balance discord. They have a lot of resources that are kept up to date by the community.
https://discord.gg/thebalanceffxiv
Egi Assault: Yes and no. They are effectively a Ruin replacement and are almost always a dps increase over using any Ruin spell, but they also provide weaving and movement opportunity. Use them often but pay attention to when you need to move, when you need to use an oGCD, and avoiding capping out the cooldown timer. Some of them have timing to worry about as well. E.g. you don't want to use Garuda's Egi Assault II until the tank has settled in on a big trash pack pull since it creates an immobile ground AoE. Using it while the pack is still being pulled would result in only hitting them for the up front damage and losing out on all the DoT damage. I also delayt the second Garuda's egi assault II so I don't overwrite the first ground AoE.
Deathflare: Yes and no. The main reason to use it early in order to line up Bahamut with the raid's buff window, in particular during the opener. It's also used early throughout the rotation to keep it lined up with it. Without going into detail, it's usually used about 2-4 GCD after starting trance. Also, if you know the boss is going to jump or combat is going to end before the end of Trance then you can use it early. That said, if you're in a pug then it is mostly pointless to try and line up buff windows beyond the first one, and there's little reason to worry about buff windows in 4-man content so I just use Deathflare at the last second to get the most movement utility as possible out of the trance.
Fester: Yes and no. Ideally you want to time these to line up raid buff windows, so they're not always used immediately. In 4-man content or in pugs, however, I personally don't bother trying to line them up outside of the opener.