Baggage Fees - UA to EI and Return
This is an ex-US trip and the DOT-modified baggage rules apply, not the standard IATA ones. Those rules state charges are based on the marketing carrier of the first segment for all legs on the ticket. So if your ticket has a UA flight number and you are sitting in BF on the outbound, those are the rules that need to be followed. If you are on a standard round trip ticket, EI is obligated to charge you based on the UA BF allowance (and cannot charge you additional at the connection point). That means they can not charge their fees for the return.
The charges for each direction should be listed on the ticket/e-receipt UA sent (at least for the first 2 bags) - there is also a baggage calculator on UA.com where you can log in and it will tell you your allowance, and you can pull up more details, based on the itin listed for both outbound and return.
CC bag fee waiver won't mean anything on this itin, as it doesn't stack with free bags you already receive (IIRC) and it only applies if you are checking in with UA, which you aren't on the return. Also, any extra allowance you might get as an elite from UA doesn't necessarily apply when checking in with another carrier. I'm not sure how that works exactly according to the IATA/DOT rules, but I've never been able to get a solid answer to that.