I think Hartnell and Troughton are underappreciated today, probably since all of their stories were filmed in black-and-white, and so many of their stories no longer exist in their entirety. Hartnell was very capable of showing us a Doctor with a sense of humor, as shown in stories such as "The Time Meddler," and Troughton was equally capable of delivering a Doctor with a tragic vibe, as hinted in stories such as "The War Games" and "Tomb of the Cybermen."
Absolutely. I've watched all of the existing episodes and the fan reconstructions of the lost episodes, and after finishing The Second Doctor's run, he became my second favorite Doctor. The Second Doctor pushed The Seventh Doctor from that spot, which is pretty remarkable considering he has held that spot since my childhood. It tears me up that so much of Troughton's work is lost, probably forever. I would love to see such stories as
The Faceless Ones,
The Power of the Daleks (my second favorite Classic Dalek story),
The Moonbase (one of my favorite Classic Cybermen stories),
The Abominable Snowmen, and
The Web of Fear (The Brigadier's first story!) in complete form. Alas. *sigh*
And not to feel slighted, but the same goes for The First Doctor to a lesser extent.
The Tenth Planet (another favorite Classic Cybermen story),
The Savages,
The Massacre of St. Bartholomew's Eve,
The Crusade,
The Reign of Terror, Nicholos Courtney's missing episodes in
The Daleks' Master Plan (I don't really care for the rest of the story

), and especially
Marco Polo are all wonderful stories that I wish I could see in completion.
In "The War Games," the War Chief, the War Lord, the Security Chief and Smythe were great villains. The dynamic between the Doctor, Jamie and Zoe was great, too. The story's huge cast kept this ten-part story from feeling stale. I've heard that many fans like this story mostly for its final episode, in which the Doctor is finally apprehended by the Time Lords, but I prefer the episodes leading up to the final one. The trial felt almost, but not quite, like it was tacked on.
The War Games is not only my favorite Second Doctor story, it's also one of my favorite
Doctor Who stories period. Some people complain about it being to long, but I think it's just right. I love
The Paths of Glory feeling of the first couple of episodes and I love the peeling of an onion effect that the story gives. With each passing episode, The Doctor discovers a villain greater than the last as he slowly learns the true nature of everything that is going on. The trial at the end is merely icing on the cake. When I first saw the serial, I thought the trial was only in the final episode, so I was pleasantly surprised when Gallifrey was featured in two whole episodes.