Checking baggage through
 

If it is all booked as one ticket then yes you will

American are not listed as a partner airline on EI website so I suspect the answer is no.

Partnership eg. OneWorld etc., is not really relevant to this possibility, whether or not they have an interline agreement is and they do:

MAY CHECK BAGGAGE TO
AA AC AF AS AY BA BE BR B6 CA CI CX DL EI
EK EY GF HR IB JL KL KQ LY MH NZ QF QR SA
SK SQ TG TK UA VS WF WS YO 9W

They can also ticket and e-ticket under interline agreements. The question arises more if they will or not even though they can. Since even OneWorld airlines are refusing to do this (check through on separate but both OW tickets), don't bet the farm on EI doing it either. But there is no lack of technical ability nor agreements preventing them.

You will need to call EI and ask about your specific circumstance, e.g. separate tickets and separate PNR's with the onward carrier AA at DUB.

EI is certainly not required to check across tickets as it would be if you were on a single ticket.

I would bet they will say no on the phone but a check-in agent might do it if there are not 1000 keystrokes required (which is likely).

Apparently it in not uncommon for other carriers (let alone alliances) to refuse to do this on two separate tickets both on the same airline, so interline or not, they may just say tough darts. If also might make a difference if the EI flight is in business class and say, refundable if someone thought it was irksome. But it won't take them long to notice Ei gets the pennies and AA gets the pounds. (Or euros, I guess).

But there is no harm asking (calling them won't amount to them arranging this for the OP) when checking in as long as both e-tkts are on paper ideally from one reservation.

I don't know why these airlines have not computed the effort - say 30 seconds - and if there is an interline agreement, adding a $20 fee at check-in, vs. refusing. Tickets on the same airline is really a particularly terrible example of putting it to the passengers, but the airlines know that at times it's cheaper to do ticket 1: A to B and ticket: 2 B to C than one ticket A to C via B and as such they are not going to encourage it.

If it's cheap enough for all this hassle, just allow for immigration, customs, luggage, check-in and security to go out and back.

People probably do this to save $50 without thinking of the effort mid-way through.

Interlining across tickets also takes time and that slows down check-in lines. There are all manner of reasons why carriers have and continue to end this service.

It can be particularly problematic for people who must clear immigration at a transfer point in order to reclaim a bag and check in and may be required to obtain a visa or may indeed not be admissible.

Fortunately, if OP is a UK, ROI, or US national, this particular circumstance will not cause an immigration issue.

General rule is same eticket, or with EI two EI tickets back to back

You can add WO (Westjet) to the list above

I've always found EI very accommodating particularly at Dublin and LHR, if you have status or a biz ticket. I had an agent try to through check a bag via Berlin with Aeroflot on a different ticket, EI once upon a time used have an agreement. I didn't even ask but they tried...