Offering multiple puzzles at the same time is a great idea, and goes well with themed suggestions. If it’s not too resource-intensive, I’d love to see that added into the mix!
Unfortunately the rest of your ideas are not that great. Offering only one puzzle from each theme is a downgrade of the current system. The first theme, the oldest puzzle, is the same as the current implementation sans the random generator. It will be the same puzzle all the time until the player solves it. The reason I left this theme in my selection is because it provides a great way to learn “in the footsteps of those who came before”.
The newest puzzle is the opposite of this. The reason I wouldn’t want to suggest this is twofold. On one hand, you can reach this puzzle with a single click of a mouse anyway. On the other hand, there’s no telling whether a player will be able to solve it just by looking at the shape. Sure, there are some exceptions, but they are a minority.
Which brings me to the longest unsolved, which suffers from the same problem. Now, I’ll put aside my prejudice against e-peen contests for a minute and try to be objective about this. The sad truth is that only a handful of people can solve these puzzles. There’s a few more who like to try, but that’s about it. For everybody else, clicking the puzzle will only yield frustration.
This problem is also prevalent in the last theme, completely random, and this has already been commented on by Eli a few posts above. In general, I would like to see puzzles suggested to the players that they can actually solve, rather than puzzles that you, me, the developers, or anybody else wants them to solve. Sure, there’s probably room for that too, but not to this extent.
Another problem with the longest unsolved would be clarification. Is the longest unsolved puzzle the one that is the oldest puzzle with 0 solvers (you can search for these anyway), or is it the puzzle that no player has solved for the longest time? If it is the latter, then right now, it would be a puzzle that has been solved by at least one player (probably more).
Granted, two of my themes are also available with but a few mouse clicks (most solvers and shortest/smallest puzzles), but they have a much higher chance of offering puzzles that a player can solve, regardless of their current skill level. For the most part, I would consider this to be good way to improve the players’ confidence, both in the suggestions, and in themselves. At least this is what my goal would be.