Religion supports science, bigtime. It's worth sacrificing a good science site to get a religion, possibly even your first site in your capitol. Arguably the AI's are so far ahead in science early on that getting a religion to help your science later on is a very good thing. There are a bunch of faith things that let you get more science, and if you have a bunch of faith you can buy the key great people (below). One Apostle sacrificed for Inquisition and a few inquisitors makes your home religion pretty safe for a long time. A strong religion is the engine that gives you strong science.
There are 3 great people you should target if at all possible. Hypatia, Newton and Einstein, spend your faith (see religion comments above) or cash to get those, the boost from them lasts for the rest of the game, run science projects if they popup. (Hildegard is nice as well).
You don't need to suze science states to get the major benefits in your buildings. 3 envoys is the key.
Most of your post is spot on, but I take issue with Hypatia (and maybe Newton).
Yes they are great, but getting those on Immortal can be very difficult, especially Hypatia.
Rushing GPP to get Hypatia early will usually come at a very heavy price, as you need to focus on campuses with buildings asap and after that most likely run research projects, at a time where you want to expand with more cities.
You might be able to buy her for faith, but that is a very big if since the AI more often than not will snag her right away as you barely got your first couple of campuses online.
Remember, this guy plays Immortal, and Hypatia on Immortal is very hard to secure consistently.
As for religion, that heavily depends on the map, and as the OP said, he wanted general advice and not circumstantial strategies (which also applies to Hypatia/Newton above).
As a general strategy, I'd say that this is bad advice.
I believe that getting a religion when going science is a lot of "ifs and buts", and not something one should do blindly as a strategy.
I finished up a science game with Gaul on Deity yesterday, and I certainly didn't need a religion for it (ended with 1300 science per turn, about turn 300 win on Epic speed).
Speaking of religion, it's even more circumstantial since if you do get one, only Work Ethic and Jesuit Education are somewhat synergistic with science, and in the case of Work Ethic, getting full benefit from it (by getting high adjacency holy sites) both delays campuses, as well as relying on particular pantheons (usually).
Jesuit Education similarly relies on a high faith income, which he might not have.
Not something I'd recommend in a general science game.
Also there is the issue of securing a religion on Immortal, which usually costs a lot of early investment into holy sites and maybe holy site prayers.
If you end up going that route, you are usually setting yourself back even further on Immortal.
It
can work, but going religion blindy for science purposes is generally not sound advice imo.
As long as he gets Kilwa though (which he should always do), suzeraining those blue city states is of top priority, and he should keep dumping envoys there just to maintain suzerainty if the AIs keep going for them.
If not, getting 1 envoy early and a bit later 3 envoys is of course a good strat (maybe 6, if it's the late game and he has plenty of research labs).