I don't think Sweden actually believed that its neutrality would survive a major war in Europe, largely because the Soviet's (and the Swedes) believed Sweden was part of the West. Although in the Twilight War Sweden's neutrality did seem to have been more or less respected.
Swedish neutrality since the early 19th Century was more to do with suiting themselves than any ideals about being a peaceful country. Sweden had pretty much fought itself out during previous European conflicts, and despite being the largest Scandinavian country it was rapidly overtaken by the larger European powers by the 19th Century. Also the other Scandinavian countries didn't entirely trust Sweden due to past dominance over them, and Sweden still had ambitions on Norway into the 1850's. During the two world wars Sweden had an uncomfortably close relationship with Germany for a neutral country, and British and French intervention in Norway in 1940 was as much to with preventing Swedish iron ore from reaching Germany as it was in helping Norwegian independence.
As Dark as already pointed out Sweden had a close relationship with the U.S. and NATO during the Cold War, and they also shared a lot of intelligence on the Soviets with NATO. They also traded weapons with many NATO countries during the Cold War, particularly with Britain and today most of the Swedish defence sector is British owned.
Sweden also believed in armed neutrality, and they were very heavily armed with a large indigenous defence sector. Sweden built and designed most of its own weapons, ranging from rifles to tanks, fighter jets and submarines. What they didn't design or build themselves they generally built under license, and license building of other Western arms and components only enhanced the already advanced Swedish defence sector.
Sweden wasn't targeted by Soviet nuclear weapons during the Twilight War, at least there is no mention of it. And it is likely to have survived conventional attack or invasion due to the fact that Sweden fighting with NATO would have severely compromised the Soviet's defence position in the Baltic and Arctic theatres due to the fact that Sweden has a very powerful army. This would have left a relatively intact Sweden in an advantageous position in Northern Europe at the end of the Twilight War.
Would Sweden have sent mercenary groups outside of its own territory with government backing to secure Swedish interests? Highly likely. In fact I would be surprised if Sweden didn't actually send larger forces outside of its territory to secure a number of Swedish interests and there are quite a few.
1) Swedes in Finland: About 10% of Finland's population is ethnically Swedish, and over 5% of them still speak Swedish as their first language. They are concentrated in the Aland Islands in the Gulf of Bothnia, and in communities on the western and southern coast of Finland. Some naval and land forces could be sent to secure these areas.
2) The Baltic Approaches: Passing through the Skagerrak and Kattegat is necessary to enter the Baltic Sea from the west. Although Sweden shares control with Denmark and Norway, Denmark geographically dominates the region. With Denmark disorientated due to the war and nuclear attacks, I could see Swedish naval and air forces patrolling this region and some Swedish land forces on the Danish island of Zealand.
3) Oil: The Swedes need oil like everyone else, and the nearest source is Norway. I think we could see some Swedish forces in Norway (maybe in cooperation with the Norwegians) to secure oil resources and infrastructure in the North Sea and Norwegian deep sea ports.
4) Southern Baltic Coast: As Sweden is still an intact nation, a lot of refugees will want to make their way to Sweden from across the Baltic Sea. So we could seen some Swedish naval and air activity along the southern Baltic coast which includes Denmark, Germany, Poland and the Baltic States. Also Sweden will be taking an interest in who's in charge and of any emerging threats across the Baltic region. So it is likely that it will be here that we will see most Swedish mercenary activity, volunteers with the unofficial approval of their government.
Army
Peacetime Establishment
Army: 44,500 (37,700 conscripts)
6 Military Commands (1st, 3rd, 12th, 13th, 15th divisions and 18th (Gotland) Brigade)
War Establishment (725,000 on mobilisation)
Field Army: 350,000
- 6 Armoured Brigade (6th, 7th, 8th, 9th, 18th, 26th)
- 1 Mechanised Brigade (10th)
- 18 Infantry Brigade (2nd, 3rd, 4th, 11th, 12th, 14th, 15th, 17th, 21st, 33rd, 38th, 41st, 42nd, 43rd, 44th, 45th, 46th, 47th)
- 5 Arctic Infantry (Norrland) Brigade (13th, 19th, 35th, 50th, 51st)
- 9 Artillery Regiment (1st, 2nd, 3rd, 7th, 8th, 9th, 22nd, 23rd, 24th)
- 1 Aviation Battalion (1st)
- 100 independent battalions (armoured, infantry, artillery, air defence)
Local Defence Units: 250,000
- 60 Independent battalions, 400-500 Independent companies
Home Guard: 125,000
- Assorted infantry, artillery, static artillery and air defence units
Equipment
Tanks: 785 (450x Centurion, 335x Strv-103)
Light Tanks: 200 (200x Ikv-91)
APC: 600 (600x Pbv-302)
SP Artillery: 30 (30x 155mm BK-1A)
Artillery: 990 (300x 155mm FH-77, 140x 150mm M-39, 550x 105mm Type-40)
Mortar: 1,500 (500x 120mm, 1,000x 81mm)
ATGW: RB-53 Bantam, RB-56 Bill, TOW
Recoilless Rifle: AT-4, Carl Gustav, Miniman
AD Guns: 714 (600x 40mm, 114x 20mm)
SAM: RBS-70, Redeye, I-Hawk
Aircraft: 19 (17x Bae Bulldog, 2x Do-217)
Helicopter: 79 (20x Bo-105CB-3, 14x Bell-204B, 19x Agusta-Bell 206, 26x Hughes 300)
Air Force
Air Force: 8,000 (5,500 conscripts)
- 4 Air Defence Districts
- 6 Fighter/Ground Attack (FGA) Squadrons
- 11 Fighter Squadrons
- 3 Reconnaissance Squadrons
- 1 Transport Squadron
Equipment
FGA: 99 (81x AJ-37 Viggen, 18x SK-37 Viggen)
Fighter: 218 (139x JA-37 Viggen, 68x J-35 Draken, 11x SK-35C Draken)
Recce: 48 (48x SH/SF-37 Viggen)
ECM: 2 (2x Caravelle ECM/ELINT)
Transport: 30 (8x C-130E/H)
Communication: 22 (3x Beech Super King Air 200, 2x Fairchild Metro III, 16x Saab 105, 1x Saab 34D)
Training: 207 (22x J-32D/E Lansen, 30x Saab 91, 105x Saab 105, 50x B.125 Bulldog)
Helicopter: 32 (10x CH-46, 12x Super Puma, 4x Bo-105CB-3, 6x Agusta-Bell 206)
Navy
Navy: 12,000 (6,300 conscripts and including Coast Defence)
- Submarines: 12 (4x Vastergotland, 3x Nacken, 5x Sjoorman)
- Missile Craft: 31 (1x Goteborg, 2x Stockholm, 16x Hugin, 12x Norrkoping)
- Patrol Craft: 11 (4x PFI, 7x PCI)
- Minelayer: 3 (1x Carlskrona, 2x Alvsborg)
- Minehunter: 23 (3x Arko, 6x Landsort, 14x other)
- Amphibious: 12 (12x LCM)
- Support Ship: 12
- Aircraft: 1 (1x C-212)
- Helicopter: 23 (14x CH-46, 9x Agusta-Bell 206)
Coast Defence: 2,650 (including 1,700 conscripts)
- 5 Coastal Artillery Regiment
- Minelayer: 16 (10x Coastal. 16x Inshore)
- Patrol Craft: 18 (18x Inshore)
- Amphibious: 154 (14x LCM, 80x LCU, 60x LCA)
- Fixed Artillery: 98 (18x 120mm, 6x 105mm, 74x 75mm)
- Anti-Ship Missile: RBS-17, RBS-15KA, RBS-08A, RB-52
- AD Guns: 40mm
Paramilitary
Coast Guard: 600
- Patrol Craft: 72 (2x Fishery Protection, 70x Inshore)
- Aircraft: 3 (1x C-212, 1x Cessna 337, 1x Cessna 402)