What allowed Swedish social democrats to hold on to power for so long?

I'd like to argue here that it was and is a combination of being able to make themselves acceptable to the old elite and co-opt popular movements that were traditionally not social democratic as well as being well-organised and well-integrated with the unions, particularly LO, the labour union.

Making themselves acceptable to the old elite

The Swedish social democrats, like many other social democrats, originally contained a very broad coalition of socialist and communist movements. However, as opposed to many other social democratic parties in Europe, the Swedish social democrats rejected an armed revolution or even a revolutionary change as the way to achieve their targets. They co-operated with the liberal party in the Swedish parliament, and the first member of parliament, the social democratic party leader Branting was elected in 1896 on a liberal party list through an electional co-operation between the social democrats and the liberals. At this time Sweden had a staggered franchise, meaning people had to have a certain income to be able to vote and a single individual could have up to 400 votes. While the second chamber/lower house had universal and equal suffrage for men from 1909, the first chamber/upper house did not, which allowed the conservatives to block liberal and social democratic laws.

While Sweden was neutral in the first world war, there was nevertheless political and social upheaval. The liberals and the social democrats were demanding universal and equal suffrage and pushing for parliamentarism, while the conservatives and the King actively rejected these demands. In February 1914 the conservatives and hte King organised a march to support increased defence spending and the King in which around 30 000 self-owning peasants (whose farms often were valued high enough that they were enfranchised) partook and were then invited to dinner at the royal castle. While the social democrats arranged a worker's march fo some 40-50 000 a few days after to support Karl Staaff's liberal government, the ensuing political crisis and conflict between Prime Minister Karl Staaff and King Gustav V resulted in the liberal government resigning and the King appointing a conservative government under Hjalmar Hammarskjöld.

The Hammarskjöld government took a strict interpretation of the Hague convention of 1907 during the first world war, arguing that a neutral power had the right to trade non-contraband goods with a warring power. Thus Sweden sold food to Germany and refused to stop, which eventually resulted in the British placing Sweden under blockade as well. Sweden was not self-sufficient in food, and in general the ideas of how much meat-based protein a human needed was over-estimated in this era. Germany thus bought so much pork that the prices increased and Swedish farmers started feeding potatoes and rye to pigs in order to make more profit, increasing the price of bread and potatoes as well.

Combined with the fact that Sweden was late and ineffective in implementing rationing and the weather conditions made a crisis rise in Spring 1917. Back in those days, potatoes were stored in stukor directly on the field - a deep groove was plowed, covered in straw, the potatoes laid in the groove and covered with more straw and then earth. Spring 1917 was very cold, and as long as there were freezing temperatures, the potatoes could not be carted to railway stations (to be railroaded to towns) and towns as they would freeze and rot on the way. The combination of increased prices, mismatched rations (there were bread to be had, but no rations and people had potato rations but no potatoes to buy) along with inspiration from the revolution in Russia caused widespread protests that often descended into riots.

In this climate, the Swedish social democrats focused on continuing to demand reformed rations, 8 hour workday, universal and equal suffrage to both chambers of the parliament and continiously told its supporters to demonstrate peacefully and focus on the Autumn 1917 election for a democratic and peaceful change. The party actually splintered over the issue, with the left-wing forming Sveriges Socialdemokratiska Vänsterparti (Sweden's social democratic left-wing party) and arguing for using the sentiments for more sweeping revolutionary change (this party would later splinter many times in communist and non-communist elements, Stalnist and non-Stalinist elements and so on, but that is another story).

The focus on democratic and peaceful change and the strong commitment to these points did a lot to ease the old elite's discomfort with the social democrats.

The conservative Minister of War, Åkerman, wanted to shake the hand of social democrat Erik Palmstierna, and he wrote (my translation):

"The War Minister, Åkerman, came to me during the debate on sobrerity and asked to shake my hand to thank us for what we had done to keep calm and order in the country.

I answered him, that he should not misunderstand the situation. Now it is calm, but next time we would neither be willing nor able to hold back. This is in your hands. The only mean to avoid a real storm would be a democratic revision of the constitution."

In the climate of the time, with revolution in Russia, Germany and Hungary, the social democrats seemed like a much, much better option to the old elite, and their rejection of revolutionary change cemented this.

The Swedish social democrats also took a suprisingly hard anti-communist stance, seeing communist movements as a threat to their own role as the single representative of socialism and the workers in Sweden. Combined with a trong skepticism of the communist revolution in Russia and Russia in general as the old arch-enemy of Sweden also made the party more palatable to the conservatives.

Continued below.

About the second question, in hindsight changing it to "what legacies did this period of social democratic rule leave for modern sweden" is more appropriate.