first off, i'll correct you in saying that using the phrase "EPA audition" is repetitious. EPA stands for Equity Principal Audition. so you are using a beginner's phrase when saying "Equity Principal Audition audition"

You can just refer to them as EPAs.
so ok... it's been a while since i've read up on the actual agreements and laws of AEA, so i hope someone will correct me if i'm remembering incorrectly.
AEA requires all producers who wish to employ AEA actors have a contract that all personnel are signed to. this is how they protect actors.
most, if not all, of those contracts ask that theatres/producers hold mandatory "public" auditions according to the contract that they operate under.
for example, all broadway shows and touring shows that operate under the "Production" contract are required to have EPAs and ECCs throughout the year. i think it's one EPA and 2 ECCs, but it could be vice versa, or completely different, i can't remember.
so often, for production contract shows, you'll see "EPA Required Call", which means that they are fulfilling their contractual agreement to hold auditions. They could be looking, they may not be looking.
some actors consider "required calls" a waste of time because the office can send anyone they want, an intern, the mail guy, an assistant, but often still, the main casting director is there, and that is why people still show up for them. they may not be looking, but it's good to get seen. often, after a lot of EPAs, the rep will leave a stack of headshots in the audition room when they leave. it's sad, but it happens.
let's say that the production contract requires one ECC a year. they can have 10 ECCs a year if they so desire, but they are only required to have that one.
and it's the same for most AEA contracts. often, theatres that operate for full seasons will have one big seasonal call for everything, and then hold separate auditions for individual shows.
so sure, they could have cast people from their seasonal audition 6 months ago, and may only have 2 roles to fill now.
let's do this for example. i do the Liason auditions in DC, they also have LOWTs. most theatres down there use those auditions to fulfill their contractual agreement for holding public auditions. and they;ll cast from their files, or agent appointments, or audition in NY, or have people that they work with all the time and use them.
which is often the case for many regional theatres that audition in NY through EPAs. they may have regulars that they want, or audition through agents, or have local auditions.
i hope i explained that ok. please someone correct me if i was wrong here. it's been a while

as for having an agent making things better... no. i see people with several broadway credits showing up to regional theatre ECCs. i have never seen a change in callback ratio whether i was represented or not.
if you aren't what they are looking for, representation- or the lack thereof- won't change their mind. and vice versa. and if you are what they are looking for, they will call you.
as for how it goes...
sign up is an hour before the calltime. at AEA, sign up starts at 830am.
the bulk of the lines show up around 730, but people have been known to show up as early as 6am, even earlier.
you get your appointment or be put on the alternate list, and come back for your appointment. you come back 20 minutes before your appt, give the monitor your appt card and headshot, and about 10 minutes of your appointment, the monitor will line your group up, and you go in one at a time. there are 7 people (or is it 5?) in a time group and you all get lined up together outside of the audition room.
if you aren't AEA, you show up whenever you want and sign the "future member" list (you can't get an appt without a signed AEA card of the right color). and then you wait. they'll usually tell you at the start of the call what will happen with future members. either you won't be seen at all, or to come back after lunch, or to hang around as FMs will be seen throughout the day as spaces in the appointment sheet remain unfilled by members.
i hope that made sense!