The release notes don't say much. The old manual doesn't say anything at all. Has anyone found a guide on the use and options for using EI and S-Log?

I think you'd have to go to the F35 manual to get that overview. But I'll pass along what I understand here, and somebody can jump in and correct me if I'm wrong.

EI mode is predicated upon the idea that:

1-You'll get the most dynamic range out of the sensor if you keep it at 800ISO

2-If you record the signal with enough fidelity (i.e. 10bit log, low compression), it doesn't matter if you add gain before or after recording/compression

So put another way, EI mode applies gain to the viewing LUTs the camera uses internally and you see while shooting, but the signal going out the dual link outputs stays at 800ISO. (No gain applied)

The offline XDCAM files will have a metadata field that specifies what EI was dialed in for viewing while shooting, for the information of the colorist. So if you dialed in 1600ISO while shooting, the XDCAM off lines will get one stop of gain applied, the signal recorded by the external recorder will get no additional gain, and the colorist will apply one stop of gain in post.

This is essentially what the Red does, the sensor lives at one ASA/ISO and all manipulations of that setting happen in the post debayer step. It works well and is handy, because the ISO setting you chose while shooting lives as metadata in the footage, and you're free to change it later.

For the F3, it seems somewhat less useful, because that metadata can only be recorded in the offline XDCAM files, which are not hard-linked to the ProRes or HDCAM files coming off your external recorder. It's up to you (or the colorist) to look at the ProRes files and go "why does this look so underexposed? Oh, I bet we EI'd to 3200" and then go confirm that by looking at the XDCAM file for that shot.

Did I make sense with any of this?

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I feel the need to explain where this info comes from (besides my owning a Red for 2 years). On RedUser there have been some very long, in-depth discussions on the pros and cons of doing gain operations in color-correction, with much of the hard info coming from Graeme Nattress. A lot of the discussion centers around just how many bits (10bit log vs 12 bit linear vs 16bit linear) you need to record the entire dynamic range of the sensor, including the noise floor.

There are those that argue that 10bit log essentially records as much dynamic range as 12bit linear (which is what the Red records as 'RAW'). In my small experience with 10bit log images from the F3 and Alexa, I'd say this is true...properly recorded Alexa and F3 images have as much room in post for manipulation as Red R3Ds.