32 married, young family - haven't declared rental income from former home

I wouldn't think the tax bill should be that high. Rent a Room should apply for the period when some of the house was let whilst the OP lived there. It's really only since 13/14 that there's an issue, at €2k a year per Brendan's calculations.

I would submit tax returns for each year and just see what happens.

I was in a similar situation with rental house, moved out, got married and was totally overwhelmed about sorting out the tax situation.
I got an accountant who acted as agent with Revenue, gave him all the information and tax wasn't all that bad. Can't remember exactly but it was offset by medical expenses and other things i hadn't claimed for. A while later i got a letter from Revenue for 2 years of Mortgage interest relief. I rang them and they let me repay in chunks over a few months. I wasnt fined anything thankfully.
It was a total weight off to get it all sorted.

Is it not a requirement that any tenancy must be registered with the PRTB to be able to claim mortgage interest relief...?

From the PRTB website;
Why is the PRTB registration so important?
Landlords must comply with the PRTB registration before they can claim mortgage interest relief on rental properties.