As long as we stay well (Stockholm is 7°/800 km/500 mi) south* of the polar circle, there will always be a sh'kia and a netz . Therefore, all zmanim can be calculated. Even a tzeis and an alos can be calculated by relying on the opinion that they occur a fixed number of minutes* (e.g. 72) after shkia and before tzeis.
Let's take some practical examples:
On shabbos korach, sunrise is at 3:31 and sunset at 10:08. This gives us until 7:34 to say Sh'ma, and sh'mone esrei has to be before 9:43. There is of course plenty of time for mincha, and ma'ariv could be said after p'lag, at 8:12, and then one could go to bed. The only real difficulty is waking up to repeat sh'ma after 11:21. Havdallah can be said then, or Sunday morning before breakfast, as havdallah over a cup of wine is only necessary before eating or drinking.
On shivah asar b'tammuz sunrise is at 3:57 and sunset at 9:50. One may of course begin fasting the night before, or take a mid-night snack. The usually lenient opinions actually come out later than 72 minutes, so eating can certainly be resumed at 11:02. Obviously, one may go to sleep in the evening, and eat in the middle of the night or the next morning.
For shabbos mikeitz, candles for chanukah and shabbos have to be lit before 2:30. Some choose to study before the meal, which also serves to push it off to a more normal hour. Shacharis can in fact be scheduled for a normal 8 o'clock the whole year, as neitz is 8:43. That shabbes is over at 4 o'clock should not pose any problems, but it is a good idea to limit the morning meal so that shalos seudos can be eaten in time. This gives an excellent opportunity to make melave malka something extra special!
The winter weekdays pose special problems, as many have to begin work before the earliest chance for shacharis. Being that the situation is temporary, one can be lenient and pray between misheyakir and neitz. Relying on opinions that specify alos by the sun's angle under the horizon, shacharis can be over around 7 o'clock. Otherwise, one would have to take a "tea" break in the morning, just as is universally done for mincha in the afternoon.
There are even more northern congregations, like Fairbanks in Alaska and Arkhangelsk in Russia, but their halachic times do not differ substantially. To my knowledge, the only permanent polar community is that of Murmansk, and I hope to add some information when I get a response from the local Chabad sh'liach.
Since we keep both opinions about tzitzis, this mitzva is not affected. Actual halachic processes in batei din are probably rare in these locations, but there are plenty of daylight hours available.
Source: Experience – this is where I come from.
* Everything here assumes that one chooses definitions so that (tzeis – sh'kia) + (netz – alos) > (netz – tzeis). E.g. if tzeis is 60 minuttes after sh'kia and alos is 72 minuttes before netz, then there must be at least 132 minutes between sh'kia and netz. This is the case for all locations well south of the polar circle.
