Do you need a winter coat?
No — a waterproof, windproof shell over a fleece or wool sweater does the job, and it's warmer than a coat in the rain. Add a light insulated jacket if you'll be on a boat, a glacier or in the highlands, where it's several degrees colder.
Shorts and T-shirts?
Icelanders wear them on the few days that pass 18 °C, and hot pools are the place to be in a swimsuit. For everything else pack long trousers, a long-sleeved layer, a hat and thin gloves.
The bright nights
Summer's real surprise is the light: around midsummer Reykjavík has 21 h 9 min of daylight and the night never gets dark. Pack a sleep mask from May to July.