Apologies for a spectacularly delayed reply. If you'd like a feeble excuse it's that we're deeply buried in significant enhancements to the bus side of the platform...
For others reading this we're talking about the live 'real-time' bus map here:
live bus map
1) Duplicate buses: Well spotted. This is was true with the 'data feed' (GTFS) we were receiving from the bus companies previously. A relatively small number of the buses were behaving as if they were carrying two separate position transmitters, transmitting slightly different locations on separate time schedules. At a guess, this is probably what was happening, so 10 points for a correct answer. By coincidence since your post we've moved over to an alternative data feed that doesn't seem to have that issue, but let us know if you spot it again. The analysis necessary to sort the data and work out where an apparent two buses were really two transponders on the same bus isn't something we can get to now as we are really busy with other stuff. Most importantly the new data feed has additional information such as the timetabled journey the bus is supposed to be following that radically improves our options in showing buses in a more meaningful way (see below).
2) The symbols aren't live, some buses appear without warning: The previous feed of position data was on a periodic 30-second batch transfer to us, so the position data was nominally 0..30 seconds old randomly for each bus. In practice there's an additional ~5-8 second delay in the bus companies sending us the data so you can add that to the delay. The new feed (SIRI) is transmitted on a more immediate basis (i.e. eliminating the stock 0..30 second delay) but there's still the delay of ~5-8 seconds before it comes to us. FWIW the data processing on our servers delays less than 0.01 seconds even when transmitting the same data to the five servers we use to process the data. Of course we are in dialogue with the data providers to ask they send the data when they get it without delay, and we're helping them improve their techniques. A bus appearing 'without warning' presumably means it's not sending it's position information. Not much we can do to detect that from our platform but gradually this will improve in part because of our work and the fact we're chivvying the industry to try a bit harder.
3) Buses take off across parks: We essentially move the bus in a straight line from point-to-point as we receive the position data, and sometimes this represents a jump as you've seen. In the prior version of the page (I think that you're referring to) we smoothly animated the bus over 30 seconds from point-to-point and with hindsight this introduced additional delay (due to the 30-second time of the animation) and made the 'park jumps' look worse. In the current version (see link above) we still move the bus point-to-point but do it in one second rather than thirty. Note at this stage the system itself has no idea of the map or the route the bus is supposed to be following, but see below.
We are working on a fundamental set of improvements in the use of the bus real-time data that have been enabled by the improved content of the 'SIRI' feed. The time/position data for the bus is pretty much unchanged, but now we can work out the timetabled journey each individual bus is on (i.e. it departed a particular stop at the start of its journey at a particular time). We have (already working) in development a high-speed 'timetable' server that given this "origin stop/time" can return the entire sequence of stops/times relevant for that particular bus (and can currently do this in 0.01 seconds). We are also working on advanced real-time analysis software that can accurately predict where on a bus journey a particular bus is, given its position and timetabled route (this is hardest part of what we're doing). This is a mix of Cambridge University research (where I work) combined with a genuine commitment to make information available to local travellers as that becomes possible.