I would go in summer, any time really. Late June to mid July is when most Swedes have their summer vacation, so it can get a little crowded (relatively speaking, still manageable in the outer archipelago). Late summer might have the better water temperatures (for mandatory skinny dipping) but the sun is up longer in late June/early July.

Weather is a gamble, it can rain a lot but the sunny days are worth the risk and getting rained on a little (and the rainy days are not a waste either). There usually aren’t a lot of mosquitos in the archipelago, but can vary from year to year. I’d say it’s never even close to the notorious areas in the north of the country.