Cycads can be removed to be transplanted at any time. In an established landscape like this one, it's best to treat them as "bare root". That is, there is no need to dig up a huge rootball and make a mess. The plant needs to be "undercut", which starts with removing the "pups" along the bottom and then carefully digging and cutting the roots. Removing the pups and doing the undercut of the roots is best left to someone with some expertise with these plants.
My post on transplanting a sago palm bare root
Timing is not important for the trimming of the leaves and the removal of the cones. It's simply a matter of aesthetics - since the plant won't be putting on new leaves for a while, you can wait to have it trimmed if you don't want it to look at a "torpedo" in your back yard.
Cycads, like sago palms, live for hundreds of years, so consider that you really don't own it, you are simply its custodian during your lifetime.