I've been to Fotografiska twice within the last year, and I do enjoy it, but it's pretty expensive, rather crowded (commercial operation, heavily marketed) and not a quick visit. They seem good about booking exhibitions that will be popular, but take a look at the website to be sure what's running next summer will appeal to you. I'm not aware of any limit on the number of tickets sold, but there's a bit of a discount for buying adult tickets online rather than at the museum. There may be a student discount that applies to your kids.
The Slussen corner of Sodermalm is still under construction, and it's by no means obvious the best way to walk to Fotografiska once you cross the bridge from Gamla Stan. Perhaps Badger or Laura can put it into words for you. I cannot; I just followed the blue dot on my phone and concentrated on getting down to the water level on Sodermalm as soon as possible. There are very limited options for crossing the road near the Slussen end, so the sooner you can find your way to the north (Fotografiska) side, the better. It's a good thing Fotografiska is open late, because getting there is likely to take more time than you anticipate.
Our sightseeing interests are pretty much diametrically opposed (I'm an art and modern-history person), so I can't help with the rest of your sightseeing plans.
If you're looking for a quick lunch one weekday on Gamla Stan, I can recommend City Sallad near the NE tip of the island. It caters to the local worker population, opening at 7 or 7:30 and closing I think at 2 PM. It has 9 salad options, all of them interesting. They'll be packaged and ready to grab out of the chilled case at opening time. They're all the same price and come with a roll and your choice of beverage (limited selection). The counter is likely to be unstaffed; you ring up your purchase on a terminal (there's only one product choice, so no complications) and tap your credit card. The cost is a very reasonable-for-Stockholm SEK 130. The machine translation of the menu sometimes lacks clarity ("ginger oumph"??), but I believe the printed version on site is better. You may be stuck with eco-friendly wooden forks; I'm not sure, because I always carry a spork in my purse in case a food opportunity presents itself.
CitySallad Gamla Stan menu
For baked goodies, Skeppsbro Bageri, a bit farther south on the same street, seems a good option. It does salads, sandwiches and coffee as well. The only thing I can personally vouch for is the cardamom buns, which are very tasty but not cheap at SEK 42.
City transit is Stockholm is very good but rather expensive (SEK 39) if you buy single tickets. I'd suggest figuring out your best option ahead of time. I used the SL smart card and loaded tickets on it, but I qualify for the senior rate and don't use transit enough to justify use of any pass. For a much longer stay last year, I scrunched all distant destinations into my last 7 days and bought a one-week pass, which was a comparatively good deal--but I bet that was at a discounted senior rate. Your situation may be different. It appears your youngest also qualifies for a discount.
Stockholm city transit