For the first problem I had, I think it was actually just a one-off error. Here's what happened: I saved my game at turn 84, creating my 19th save, and then exited the game. Later, I loaded that 19th save and played for awhile, creating my 20th save and leaving again. Then, when I loaded the 20th save, I was back at turn 84, where my 19th save had been. I played for a few turns, created save 21, and left. Then I came back and loaded the 21st save, and once again I was at turn 84. This pattern continued, with each new saved game always putting me back at the 84th turn when loaded.
If you notice with that progression, I was always loading my newest save. I think some sort of error occurred the first time I loaded the 19th save. Then, this error was passed on to each subsequent save. I solved this simply by going back and loading my 19th save again, ignoring the 20th, 21st, etc. Now, I've continued my game awhile, through multiple saves and loads, and everything is fine. Weird!
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The other problem with getting a game to load is a different issue. When I first return to the site and load a game, it asks me to choose a player name and login. The name I use, Archimedes, is automatically in the field, so I just hit the Login button. I then get a message saying "Loading game failed", and an OK button. This happens every time I first return to the site after previously closing Firefox. I then get the "Welcome to Freeciv-web" window and I choose Start Game, which leads to a new game being created. I haven't found any way to get around this bug.
After ending the new game, I attempt to load one of my saves again, and this time I do not have to choose a player name. It goes directly into the game and I get the "Welcome to Freeciv-web" window, where I again choose Start Game. However, if I click the Start Game button too quickly, then it will create a new game and, with the list of civs in red text, it says
"Archimedes lost control cmdlevel on game start. Use voting from now on."
This is what was happening to me before, forcing me to make a dozen attempts sometimes before successfully loading a game save. Now, I've figured out that if I simply wait at the "Welcome to Freeciv-web" window for a few seconds until I see the game loading in the background, and then hit the Start Game button, it loads successfully every time.