Checked luggage w/ multiple airlines

You are subject to the baggage rules of the airline who is checking you in for that flight. I would try to comply with the airline with the most strict requirements. Checked luggage would not be problem assuming your checked luggage is fairly standard. It is just what is allowed as carry on.

It is hard to answer since no one knows what kind of ticket you bought.
You have to read the details of the trip confirmation all the way through.

It is a new one to me that American Airline sells tickets or are code share partners with Aer Lingus and United when it is not part of the Star Alliance

Will I be subject to different policies based on each leg of the flight

Yes

Whatever name is painted on the side of the plane you are boarding, is who sets the baggage rules.

And rules for flights within Europe may be different than those for trans-Atlantic flights, even on the same airline.

As stated previously, every airline has their own rules.

Here is a link to Aer Lingus' baggage policy. If you have to pay, and you might depending on the fare you have, it is best to pay ahead of time than at the airport to save money.

https://www.aerlingus.com/travel-information/baggage-information/checked-baggage/#/tab-0-flights-within-europe

Also check your fares for United and American as some of their lower fares don't include checked luggage.

A suggestion, carry on a big enough bag to hold things you wouldn't want to go amiss (your camera, a change of clothes, maybe the formal wear, a toothbrush). That way, even if you have a snafu, you end up at your destination with things you need for a few days while waiting or luggage to find you.
Also, you can pack a soft/collapsable duffle into your smaller luggage for things you want to bring back from Europe. In other words, dont pack for the departure in anticipation of the size you'll need for the return, just bring an extra duffel (that you can check on the way back and that can get lost, worst case scenario, and still find you at home eventually). Sounds like the only extra you'll have is the camera and formal wear. I bet you can fit all that in a carry on if you want

Then follow what everyone above said :)

My interary includes:

St. Louis to Newark (United)

Newark to Dublin (Aer Lingus)

Dublin to Vienna (Aer Lingus)

Berlin to Dublin (Aer Lingus)

Dublin to Chicago-O'Hare (Aer Lingus)

Chicago to St. Louis (American)

Will I be subject to different policies based on each leg of the
flight, or will my American ticket get me through?

Yes, you will be subjected to different policies depending upon which airline you purchased the tickets. You also subject yourself to the greater likelihood that your checked bag could get lost.

EU rules are MUCH more generous about compensation for lost luggage than flights under US rules. For example, if United & American lose your luggage, you will get the minimum in compensation because those legs are both domestic.

American Airlines ticket/confirmation should have included a baggage policy describing checked luggage conditions and fees. Note American's code share policy: https://www.aa.com/i18n/travel-info/partner-airlines/codeshare.jsp?anchorLocation=DirectURL&title=codeshare

Our codeshare agreements make booking, checking in, checking your bags
and traveling more convenient. Plus, the coordinated schedules make
sure you'll have enough time for all your connections.

Check in with the airline operating your first flight

Baggage policies differ between airlines and other charges may apply;
check with the airline operating your flight for baggage information

My last (2018) American Airlines flight was a AAdvantage Milage program award that American fulfilled with Alaska and Horizon Airlnes flights. Although a "free" ticket, Alaska baggage fee policies appied. (RT from Medford to MSP cost $11.20 for taxes and fees plus Alaska's baggage charges). My ticket included baggage information reading:

Baggage charges for your itinerary will be governed by Alaska Airlines
BAG ALLOWANCE -MFRMSP-No free checked bags/ Alaska Airlines 1STCHECKED
BAG FEE-MFRMSP-USD25.00/ Alaska Airlines /UP TO 50 POUNDS/23 KILOGRAMS
AND UP TO 62 LINEAR INCHES/158 LINEAR CENTIMETERS** 2NDCHECKED BAG
FEE-MFRMSP-USD25.00.... Etc etc etc

American Airlines ticket/confirmation should have included a baggage
policy describing checked luggage conditions and fees. Note American's
code share policy:

The American airline ticket is all by itself at the end of the trip from ORD to STL. Aer Lingus' code share is with United so AA's codeshare policy isn't relevant to the OP. That AA domestic flight is connected with nothing on the OP's itinerary.