It isn't the midge bites that are the problem. It's the ticks that may infect you with borrelia and TBE (Tick-Borne Encephalitis). The latter may have some really nasty side-effects. Pretty much all of the area around lake Malaren and the Stockholm archipelago have a lot of ticks.
Sweden isn't cheap by any means (none of the Scandinavian countries are). Until recently, the GBP was very strong against the SEK with about 14 SEK to £1, but that is more like 10 SEK to £1 now. Eating out is generally expensive. Alcohol is very expensive and places like Grona Lund and Skansen are expensive too. But then again, so is a day at Alton Towers here, so I guess it depends on what you compare with. If you choose to self-cater it is worth bearing in mind that VAT is charged on groceries in Sweden, so those prices are higher than here as well.
Skansen is like a miniture Sweden and well worth a visit. They do have a zoo part with most of the fauna unique to Sweden, including moose and wolf and, if memory serves me, lynx and wolverine. I'm less partial to Grona Lund - I prefer Liseberg in Gothenburg for that sort of thing - but that's just me.
The one thing you should not miss in Stockholm though is the Wasa Museum. It's the Swedish equivalent of the Mary Rose, but so, so, so much better. Basically, apart from the masts the entire ship has been preserved. When I visited they had reconstructed the masts and rigged her. Really impressive. Lots of hands on stuff for kids as well. The Mary Rose museum in Portmouth seemed like nothing after that.
Other things worth visiting are:
- Djurgarden as a whole, where Skansen, Grona Lund and and the Wasa museum are all located is an attaction in itself.
- Gamla Stan (the Old Town)
- Fjallgatan (a street on the Southern island that overlooks the inlet into Stockholm and the Finland ferries. There is a cafe with tables along the edge that serves really nice coffees, cakes, light lunches and ice-cream. It is very popular with the Stockholmers though so it may be very busy. But even if you don't get a seat, the view is spectacular and the area around it is interesting to explore.
- Drottingholm (the royal residence)
- Fafangan (another cafe on top of a hill by the water which is very popular with Stockholmers. There is also a restaurant there.)
- Millesgarden (former residence of Carl Milles, sculptor and artist. THe garden is filled with sculptures)
-Then there are of course the royal castle, the cathedral, etc.
Stockholm is a very beautiful city, especially in the summer. It is comparitively small so the centre is easy to cover on foot. There are a lot of cycle paths so cycling in the city is fairly easy.) The busses and underground (T-bana) are clean, safe and generally reliable and you can get transport passes that give unlimited access to travel transport and reduced entrance fees to many attractions.
If you want to explore the Stockholm archipelago, island hopping and staying at campsites or hostels is quite popular. You can also do one day excursions.
If you want to eat cray fish, you'll have to go around or after the cray fish premiere which is on the 8th August or the first Wednesday in August, depending on different traditions. By that time, the Midnight Sun is no more at least not in the Southern half of the country.
If you want to dance around a midsummer pole, you'll have to be there for Midsummer's Eve which is celebrated on the Friday following the summer solstice. Unless you work in the emergency services or health care, or similar, NOBODY works that day and EVERYTHING is closed, including any restaurants that aren't along the most tourist-intensive streets in the big cities. I'm not saying that you shouldn't go around that time, just that you need to be prepared. Of course, the hotels and their restaurants would operate as normal.
If you want to try something different from Stockholm, Gothenburg is very nice. It has a completely different feel than Stockholm but a really interesting place to go. Another place that is lovely is Gotland, an island in the gulf of Botnia. It's beautiful, with lots of history and lots to explore. All the hills on Gotland are concentrated in the main town, Visby, so it is very good for exploring by bicycle.
Sweden is a child friendly country to travel in but there may be some language problems. Not because Swedes don't speak English (they do and very enthusiastically and well in general), but because things aimed at children will primarily be in Swedish and may not be fully translated. I think however as long as you stick to the normal touristy things, this shouldn't be a problem.
Cheapest option for accommodation is staying at "youth" hostels (open to everybody)that are run by the Swedish tourist association, STF. They are generally clean but you have to bring your own sheets and towels, although many do rent them out so it is worth checking with them before you go. Most also offer breakfast at an extra cost, at least during the summer months. Again, check in advance.
HTH. Sorry for long post 
.