Most of these questions seem extremely hard if not impossible to answer without something like talking to the makers of the sim or finding hacks to get at the source code. Some of these are however slightly testable. For example, if it turns out that BQP is comparatively large (to fix an example say it contains NP), then it would suggest that if we are in a simulation then the simulation is not being run on a classical computer.
One possible test to see if we are in a simulation, although a very weak one, is to collect a very large set of data about decaying particles and then analyze the data. If the data turns out to be much closer to a pseudorandom rather than truly random sequence then this would strongly suggest we are in a simulation. Note that if presently believed complexity conjectures are true, distinguishing actual pseudorandom data from genuinely random data should be extremely tough.
One serious issue about engaging in tests to try to see if we are in a simulation is that the most obvious tests are often tests which have some danger of potentially crashing the simulation or causing the programmers to stop it. Thus for example, one test that jimrandomh mentioned to me was bouncing laser beams off of farther and farther objects to expand the radius which we know is being accurately simulated (this mainly is usable for ruling out a simulation that only simulates our solar system in detail). This is however exactly the sort of thing which one could see as leading to the equivalent of a seg fault or other error. On the other hand, if the simulators are only running a simulation on our solar system then they are more likely to care about humans or life in general on Earth (at minimum, we're then a much larger and more significant part of the simulation) and so are more likely to have anticipated this sort of problem and made sure the code is somewhat resistant. That said, if all probes stopped transmitting back to Earth or showing any signs of existence a bit after our solar system (say 75 AU out) and all attempts to bounce lasers or radar off of distance objects failed, that would strongly suggest we are in a simulation that just includes our solar system.
Xixidu tweets about experiments likely to crash the sim quite often; for example: http://www.centauri-dreams.org/?p=18718&utm_source=rss&utm_medium=rss&utm_campaign=spacetime-beyond-the-planck-scale
And I somehow feel like I've read a SF short story long ago where someone found out radioactive decay followed a simple linear congruential generator, or something.
My father, who is home recovering from surgery, emailed the following web page to me and a few other members of my family, and expressed interest in reading interesting responses.
Any thoughts?