Is EI a valid carrier on a OWE?

No. EI is not a oneworld airline.
Rule 3015

EI is allowed on a Global Explorer ticket, which is what rob_88 has been talking about.

Yep, this:

Seems it successfully ticketed overnight, and looks like it's all DGLOB34, even including one Eco (B class) sector on BA.

It's BA, EI, AA, JL, and QR. Ticketed on AA stock, booked through Bookwithmatrix. And it came in cheaper than my Corp TA could price it, so I'm pretty pleased. I'm getting where I need to go, with a scenic detour to see some friends on my way home, and it's gives me 3x what the equivalent BA itinerary would give me in their new program.

It's a total ballache getting Matrix to price a DGLOB34, but with a bit of patience I got there. Also looks like MatrixPowertools wasn't able to push my booking into the AA flow, while Bookwithmatrix was - though it only supports the US AA site. So, if you're okay paying in USD and your bank isn't overzealous with fraud detection then this is viable I think.

it would be cool if xDONEx could be done on same way

Several questions arise.
How many segments? Most online websites have a limit of 8 segments.
Where does the itinerary start if it is not a low cost origination?
Was the price converted to USD or the USA base fare?
Are you okay with sharing the itinerary as it may meet your needs but not the needs of most posters in this forum?

Sure:
Osl-lhr on BA
Surface segment
MUC-DUB-SEA on EI
SEA-LAX on AA to trigger AA selling eligibility, otherwise I'd have done JL Codeshare
LAX-NRT-SIN on JL
SIN-DOH-MUC on QR
MUC-LHR-OSL on BA, but can take my pick from available Codeshares to tee up my next one ex OSL.

NOK converted to USD, albeit at an unfavourable FX rate due to the Commander in Cheeto's nonsense.

It would be cool if the tool worked reliably.

We are trying to book LHR-GIG-EZE as part of a RTW trip in April using our travel agent. LHR-GIG is fine, but so far no availability on GIG-EZE. This is the BA flight that goes from LHR-EZE with a stopover in Rio. Has anyone tried to book GIG-EZE separately and if so did you have any issues with availability (or does anyone know what the issue might be here?).

Depends on the date and fare class. You didn't mention whether you're in business or economy, give us an exact date, or tell us how many seats you need, but, for example, here's what I see on EF for April 2-8 (there's only one flight per day, BA 249, a 777 with no F cabin):

April 2 -- D6 L0
April 3 -- D0 L9
April 4 -- D1 L9
April 5 -- D6 L0
April 6 -- D8 L9
April 7 -- D1 L6
April 8 -- D0 L7

So it appears, at least to me, that your problem is likely a simple lack of availability in the cabin you've booked on the date(s) that you want.

To add to Dr. HFH 's information you don't indicate whether you want a stopover or a transit at GIG. Availability can vary depending on that, and, also, it can vary based on the originating country of your xONEx.

It is a stopover, we are looking at on a DONEx, actually 24th of March. EF is showing D0, we have been trying for a few weeks. Might ask our agent to look at a day earlier as it is showing D8.

EDIT: originating in OSL and first flight OSL-LHR

Also take into consideration any special events happening in these cities. You are flying just a few days before Palm Sunday.

In that case, I'd lock the day prior in.

It seems BA feel they do not need to offer discount business fares on that date at this stage. This can change with D availability appearing at a later date.

If it does, then since it is a stopover, it should be a strait forward revalidation to change to your preferred day.

A bit OT, do we need a sticky for actual bookings and cost? Maybe following some format that includes dates, how booked, and airlines? The information here that includes the ins and outs of how to book seems to obscure actual examples of bookings.

Actual examples really do help newbies and everyone else.