1.) Does the solicitor they use for the conveyancing have to be based in Ireland or can they use a solicitor in New Zealand?
Technically, they could use a NZ solicitor but its unlikely they would know the Irish system well enough to do the work properly. They should use an Irish solicitor - bit of an issue though with money laundering issues and getting paperwork signed unless they plan to come to Ireland for the duration of the purchase. Also if it is an Irish mortgage, bank will insist on Irish solicitor.
2) Dont know
3) all the usual costs. Fees, VAT, outlays, stamp duty.
mf