Who was the last king to take part in combat while they reigned?

I can answer the second-to-last, and last question:

Did other reigning monarchs fight in major battles during the 1600's?

I assume we are restricting the question to Europe/the West. Even then, the answer is definitely yes: I can think of several off the top of my head, and there are likely many more that I cannot:

Gustavus Adolphus of Sweden fought in many battles, and was killed during a cavalry charge in 1632 at the Battle of Lützen.
Maurice, Prince of Orange, was active during the Thirty Years' and Eighty Years' Wars; while not a king per se, he was Prince of Orange.
King Christian IV of Denmark fought or commanded personally in various wars, both on land and at sea, including the Thirty Year's War and the Torstenson War. Several contemporary paintings show him having lost an eye in combat.
Charles I fought during the English Civil War, of course, though whether he actually saw real action personally is unlikely (beyond the executioner's blade). And, of course, Cromwell, who became Lord Protector of England, a sort of king, was most certainly an active combatant in many wars both leading up to and during his Protectorate.
Slightly outside your time frame, but Charles XII of Sweden is generally thought to be the last European king to actually die in battle, in 1718. Charles was an absolutely brilliant military commander, who was better suited to generaling than ruling and fought in many battles personally. He was killed in somewhat mysterious circumstances during a siege he was commanding, when he was shot in the head.

Who was the last king to take part in combat while on the throne?

This has been asked several times in this subreddit, the below summarizes my previous answers, both of which center on Napoleon III of France, and a bonus. Again, assuming here that you're interested in the West.

The Battle of Sedan, which ended the Franco-Prussian war (for all intents and purposes), is generally considered the last time in which a reigning monarch took part in actual combat (at least in the West). Emperor Napoleon III was in direct command of French forces as they fought against a numerically and qualitatively superior German force, and were decisively beaten. There are eyewitness accounts of the Emperor intentionally running into battle and attempting to expose himself to enemy fire in the hope of being killed, so that he would not have to suffer the humiliation of personally surrendering -- at least according to Emile Zola.
The Battle of Solferino, during the Second Italian War of Independence, was the last time in which both armies were directly commanded by monarchs. Again, Napoleon III at the head of the French army, and Emperor Franz Josef I at the head of the Austro-Hungarians, as well as Victor-Emmanuel II, King of Sardinia (and eventually Italy). While these men were in direct command of their troops, the degree to which they actively participated in combat itself is obscure and debatable. It is known, however, that Franz Josef and especially Napoleon III were greatly affected by witnessing the mass suffering of wounded soldiers after the battle, and Franz Josef personally stepped down from command after this, so they were in among the carnage fairly closely, at least for a time.
While not a king, Francisco Solano Lopez of Paraguay was a monarch in the sense of dictator and for all practical purposes was a king/emperor, though his title was President. He was killed while fighting in battle (technically the battle was over and he had decided to fight to the death against a contingent of soldiers sent to disarm him) during the War of the Triple Alliance in 1870.