Free travel card and Dublin airport

Generally, yes. It can be route dependent with some operators. Do check their websites or contact them.

Note that some long distance operators route their services to the airport before the city centre, whereas others do it the other way around.

Yep, had to check this for my Mother the other day. As long as you can show your pass that's fine. Again Google is your friend. I think it's listed under the Accessibility heading on the Luas terms and conditions.

You cannot use the Free Travel Card with Dublin Express https://www.dublinexpress.ie/

Dublin Express is a private service and the only service from Heuston Stationto the Airport. Dublin's Buses now permanently ceased Airlink 747 and 757 service was also not available on the Free Travel Pass.

Best option is Luas to Abbey Street and then either the Air Coach 700 or Dublin Bus 16 from Upper O'Connell street or the Dublin Bus 41 from the closer by bus stop.

Thanks.