A standard Rabbit card has a validity period of five (5) years, and an inactivity period of two (2) years.

Not sure what is meant by "conversion"? Maybe from an old BTS stored-value card to a new Rabbit card? Not sure they've been around long enough to be invalid yet?

So you need to convert the MRT card to a new one by June 2016, but this is still not the spider card??

Yes, and yes.

Does this mean yet another conversion, or will Rabbit, new MRT and Spider all work together

Who knows, but probably not. Rabbit card for BTS, MRT card for MRT. Spider card, not holding my breath.