I am aware that they had excellent pikemen, but literally every other nation before them had spear troops too. Were the Swiss pikemen capable of aggressive maneuvers that other spearmen couldn't do? I heard somewhere that their pike divisions were often mixed with Zweihanders, how did this work? In what major battles did Swiss troops prove invaluable?
Were the Swiss pikemen capable of aggressive maneuvers that other spearmen couldn't do?
Yup. The Swiss companies were disciplined and well drilled to do things other formations couldn't. By and large at this point infantry was not very mobile.
Before the 15th century it was still very much a variation of the hundreds year old spear/shield wall. Pack a bunch of people together and they stay behind their shields (and later their plate armor.) First side to get tired and start to break up either loses or gets charged by the mounted knights to help break them then kill/capture them.
Fighting Techniques of the Medieval World: Equipment, Combat Skills and Tactics By Matthew Bennett Starts at page 8.
What the Swiss did was take the spear wall and divide it into 10x10 squares of 100 men. Even on all sides to change direction quickly enough to avoid being attacked on an unprepared side. This was a must considering their ace up the sleeve, the pike charge.
See, other countries could field pikemen, but they couldn't field a force coordinated enough to march in formation into an enemy. The regular pike wall was largely static. You gather up your men, pack them together and tell them to point the point at the enemy. This is a very intimidating (read impenetrable) sight to cavalry, who generally won't ride into the teeth of death.
So the Swiss could take a cavalry charge proof force of infantry, and march it around a battlefield. What do you do with 100 people armed with long pointy sticks? March them into the enemy that's what. The Swiss were very good at this. There was no counter to a well ordered Swiss square charging into your troops. It was responsive enough to be very difficult to outflank and very deadly from the front.
I heard that their pike divisions were often mixed with zweihanders, how did this work?
So they probably didn't use zweihanders in their pike formations. It's not impossible just unlikely. What they would have had was halberdiers, a much more effective pole arm. Almost as long as a pike, but complete with an ax blade, a hook to pull down riders and maybe a spike to kill people in armor. If they did have zweihander they'd have used that the same way, it just would have inferior reach. Zweihanders are really less swords and more pole arms in terms of how you use them, but they're also status symbols, often used by bodyguards.
Last point, battles in which the Swiss earned their money.
Burgundian wars the Swiss were hired by the French to fight the Burgundians. The Franco-Swiss forces win hands down and kill the duke of burgundy at the battle of Nancy. The Swiss companies outflank duke Charles and charge down a hill at him, meaning his primitive cannons couldn't elevate high enough to hit the Swiss. They're said to have fired one volley and killed two people. The duke was killed by a blow to the head from a Swiss halberd.
The Italian wars 60 years of fighting between almost every major European power to control Italy. The Swiss served largely for the French. By this time the Holy Roman Empire (confederation of German states) had started to raise its own pike formations to rival the Swiss. These were the Landsknechts. They maneuvered and even dressed like the Swiss, although they were generally not considered as good. They hated each other. The Italians, who approached war much less viciously, called the fights between the groups "bad war." This was "tak eno prisoners, till the bitter end" type rivalry.
Battle of Novara The French beseige an Italian city defended by Swiss. A Swiss relief force encircles them and beats them. The Swiss massacre the Landsknechts hired by the French, 5,000 dead out of 10,000. Unheard of casualty rates for Italians, who prefer to change sides rather than fight. (Only a slight exaggeration.)
Edit: Source: Starts at page 8 of the PDF linked below. (Adding more sources above for my first few paragraphs to back up the assertions there since my original source doesn't really cover that in depth.)
The AskHistorians Podcast: http://askhistorians.libsyn.com/askhistorians-podcast-037-war-and-politics-in-the-long-18th-century Delves into significant detail on why swedish pikemen were successful in this era. Are Swiss successes related to this? If not, can someone explain why?