So our journey is

MAN (Manchester UK) - ATH (Athens) - Easyjet

5 hour Layover

ATH (Athens) - AUH (Abu Dhabi) - Wizzair

There are 3 of us on this trip (myself, wife, daughter) and only checking in 1 fullsize suitcase.

so my question is, when we land at Athens, we will already be airside, so I thought it would make sense that my wife and daughter just stay airside, find the gate for our next flight and just wait there.

whilst I go check-out, collect luggage, and basically re-check-in the luggage for the ATH-AUH flight and then obviously go through security again and then meet them at wherever our gate is for the flight.

It makes sense for just 1 of us to do this instead of all 3, so I was just wondering because we are getting 2 different flights with 2 different airlines. Do we 'all' have to check-out and back in again or is it OK for just me to do it?

Thank you in advance for any advice

1 person getting 3 boarding pass's may be an issue.
International flights can require a document check.

Unless you will be able to generte the boarding pass in the Wizzair app or website, all 3 would need to pick-up the boarding passes at the check-in counters. 1 person can't get boarding passes for the 2 others. For a flight to AUH I think there is a high chance that a document check will be required at the check-in counters and you can't generate boarding pass yourself (but admittedly no first hand experience).

Do transit passengers traveling on two separate tickets have to go through EU arrival processing upon landing at ATH, and EU departure processing upon departing for AUH?

If you have a BP for the next flight I would expect you to be able to do non-Schengen/non-Schengen transfer, i.e. you remain airside without entering EU/Schengen (but no first hand experience with ATH). If no BP you need to enter EU/Schengen and proceed to the check-in counters landside. If you arrive from EU/Schengen you obviously need to go through Schengen emigration/passport control before being able to board a flight to non-Schengen country.