The mould thing is tricky. I had a mouldy pillowcase and tried a salt solution, scrubbing with sea salt, soaking in Napisan and dabbing the mould with clove oil and none of that worked. I just kept chucking it in the wash and hanging it out in the sun every time I did a white load and eventually the mould washed out (but it would have been at least 15 washes...probably more).

Given the age of the bunny rug that may not be ideal. I would patch test for colour-fastness with an Exit Mould type of product and if it was all good leave it in for just a couple of minutes on the mouldy marks and then rinse and get it in the wash asap.

As for the other stains...I would still try a normal wash and sun dry first so you can see what you are dealing with...maybe spraying a bit of pre-wash stain remover first. Then go for the soaking option if you have to.