It depends on what you mean by "fought."

The last major world battle in which armies were under the personal command of their kings/monarchs was the Battle of Solferino, which was essentially the final battle in the Second Italian War of Independence.

At Solferino, a huge army of Frenchmen and Sardinians of about 140,000 was under the direct command Emperor Napoleon III and Victor-Emmanuel II of Sardinia. That army fought against an army of about equal size under Emperor Franz-Joseph I of Austria, making Solferino the largest battle since the Battle of the Nations.

However, neither of the three leaders actually crossed swords, so to speak, with one another directly. I'm afraid I don't know the last battle in which heads of state actually fought hand to hand with each other, but I will say such a practice was, to be sure, very uncommon. Kings were extremely valuable for a variety of reasons, and, even in Medieval times (and before), tended to be surrounded by bodyguards on the battlefield who would protect them from would-be challengers.

Returning to Solferino for a moment, it's worth mentioning the reasons that this was the last battle in which monarchs commanded their troops. The battle was enormous, and featured an early glimpse at some of the kinds of casualties and carnage that would later become commonplace in the Civil War, and for some of the same reasons. In the aftermath of the battle, all three monarchs, but especially Emperor Napoleon, were strongly affected by the sight of the many wounded and suffering soldiers. Napoleon III actually decided to put an end to the whole war as a result of this, and a third-party observer of the aftermath of the battle, called Jean-Henri Dunant was so moved by what he saw that he started a campaign that eventually led to the establishment of both the Geneva conventions and the International Red Cross.

There is a famous photo of Salvador Allende, President of Chile, wielding a machine gun right before the attack on the presidential palace by the golpist in 1973. The golpe was successful, and Pinochet become the dictator of Chile. Allende was slain in the attack. It is not clear if he actually fought.

Probably best to drop that "inbred" unless you can substantiate it. The Habsburgs had a major problem with inbreeding, not royalty in general.

You are also assuming that recent royals are ornamental. You might like to look at the career of Lord Mountbatten as a counterexample. More recent members of the British royal family have not been as notable, but nevertheless it is their policy that they serve in the armed forces, and if necessary on the front line. Prince Andrew, for instance served in the Falklands war as a helicopter pilot. The small note in Wikipedia about "Exocet decoy" refers to a technique whereby the pilot places his helicopter in front of an anti-ship missile, some distance from the targeted ship, with the intention that the missile will be confused and aim for the midpoint between the ship and the helicopter. The technique carries some risk.

Subsidiary question: when was the last time that the children of a serving president of the USA served on the front line?

You may be interested in the 'Did leaders participate in battles?' section of the Frequently Asked Questions pages, as found in our sidebar.