Harpa Concert Hall, Braud & Co, Reykjavík Röst. Uppsalir, Café Babalu. And finally a stop at the local supermarket (Bonus) to buy some skyr, the local Icelandic dessert, rich in protein.
Winter: between 12 to 4PM
Summer: Morning or evening are the easiest time
Iceland was rated one of the 5 safest country in the world, therefore you do not have to worry in terms of safety. The only thing I would warn you about is ice: in winter, streets can become very icy, be careful when waking down the streets and if needed, stop at a local grocery store, they all sell small city crampons to wear over your shoes.