Visiting last week of June

Visiting Iceland the last week of June with my husband and our 18 month old and a group of others - will be doing mostly the touristy things, but trying to prep and buy what we need to pack now. What will our necessities to wear be for that time of year? Any tips for visiting with a toddler? And would open-toed shoes for a more formal event be totally out of the question, weather wise? Not entirely sure what to expect. Also - is the blue lagoon strict on the age minimum of 2 years? Thanks!