Updated : 24/05/2017
YES. Duty-free shopping, for example in airports, is only for non-EU residents whose are travelling to a destination outside the EU. EU residents have to pay VAT, even when they shop in another EU country.
You may sometimes be exempt from paying VAT when you buy a car in another EU country.
YES. As long as you present the goods to customs on departure from the EU, together with the VAT refund documents, within 3 months of buying them.
YES. You have to prove you are resident outside the EU and you have to export the goods from the EU within 3 months of buying them. In some countries there is also a minimum value of purchases to qualify for VAT refunds, but this is not the case in Ireland.
NO. When you're leaving the EU, there are no limits on the quantity or value of the duty/tax-free goods you can buy.
However, there may be a limit on what you can bring in to the country of destination (in this case the USA). You need to check this with the (US) authorities.