1. I live in Stockholm Sweden and here in summer time i.e. during the months of June to August the day and night timings are very odd. The sun sets at 11pm and then rises again around 2 am. So the time is very short and for the people who work and go on jobs it is very difficult to stay late to pray fajr prayer. So please can you tell us how can we solve this problem of fajr prayer?
2. The timing of Fasting in Sweden is around 18 hours in summer. Its very difficult to fast for that long. Some people open the fast according to the timings in Pakistan or Saudi Arabia. Is it right to do so and what timings should we follow?
The time of prayers and fasting was set based on the criterion most suitable for the people in the Arabian Peninsula and areas around. Accordingly people used to pray Fajr as the first thing after waking up and before starting the date. Also people used to fast for a length of time each day that was reasonably within their capability and with little variation every year (In Mecca the shortest fasting time is about 12 hours while the longest fasting time is about 15 hours, that is, an average of about 13.5 hours per day and a variation of no more than about 3 hours).
These criteria are easily adoptable and suitable for most of the places around the globe, there are however some places nearer the poles (like your place) where using this criteria might result in more than normal difficulty in everyday life (in some of the Scandinavian countries for instance the fasting time can get as long as 21 hours during summer and as short as 9 hours during winter, meaning the variation of the length of fasting throughout years will be about 12 hours). Similar conditions are true for the Fajr prayer.
First I need to emphasize that the solution to this problem cannot come directly from Sharia (religious law). Obviously this was not an issue at the time of the Prophet (pbuh) and not even at the time of the companions. This is therefore a matter of Ijtihad and personal opinion and people can have different views on this.
Appreciating the above, this is my view about each of the two issues you raised:
1. Fajr prayer:
The obligation is to wake up for Fajr prayer. We usually need to go to sleep with full intention to wake up for Fajr prayer. We also normally use some external measures to make sure we will not miss the Fajr prayer (like an alarm clock or asking a friend or a member of family to wake us up or staying up till Fajr if it is close to midnight). These external measures however are not an obligation although using them is highly desirable. In other words, assuming that we have full intention to wake up for Fajr prayer, it is not our religious obligation to set up alarm clock or to ask a person to wake us up or to stay up till Fajr. We can go to bed intending and determined to offer the Fajr prayer.
In this situation, if we happen to miss our Fajr, this will not be a sin because we did not miss it intentionally. However as soon as we wake up we need to fulfill this obligation and offer it.
The implication of this for the particular situation you mentioned is that during the summer time when you need to go to work and you find it difficult to wake up for Fajr, you may go to sleep without using any means to wake you up for Fajr but with full intention to actually wake up for it. If you wake up on time you will do your Fajr and if you do not wake up on time and the time is due, then you will do your Fajr as soon as you are up. This way you are with all honesty putting your trust on your Lord, believing that if you were too tired to actually wake up then your Lord forgives you and accepts the prayer that you will offer as soon as you wake up, and if you were not that tired your Lord allows you to wake up and read your Fajr prayer on time.
Please note that the above is only to deal with the exceptional circumstances that you described (odd times, working hours, etc.) otherwise in less exceptional circumstances the most desirable is that one needs to take benefit of every means possible to wake up for Fajr.
Please also note that you may find some preparatory measures to be helpful for waking up for Fajr (like a nap during the day or at late afternoon, having a light dinner, etc.) however using these measure too is not an obligation.
2. Fasting during long hours.
There can be two opinions on how to deal with this problem.
One opinion, as you mentioned, is to adopt the fasting hours of a country that has normal hours of day and night. The question of which country that needs to be can never enjoy a definite and agreed upon answer because as I wrote above the whole thing is based on Ijtihad and a matter of opinion. Given the general spirit of being in line with the community and doing things collectively, in my view the best is that (if we go with this solution) we follow the majority of the Muslims around in deciding which country to consider as the base for fasting hours.
Second opinion is that as the Qur'an implies we need to fast if it is within our normal capability and if it is not then we can simply make up the missed fasts during other times of the year.
The answer to the question: "Which of the above two solutions to follow" again does not have a definite answer because it is a matter of opinion. Again, where possible, in my view it is better to follow what the majority of Muslims around us are doing.