Questions
How securely does it need to be fixed?
Properly. A cat lands in this at speed, so fix into studs or use rated wall anchors. Drywall alone will not hold it and this is not the place to improvise.
Will my cat actually use it?
If it already jumps to that height, almost certainly. If your cat stays on the floor by choice, a raised bed will not change that.
Can I put it in the washing machine?
No. Spot clean the plush lining; the moulded shell is not washable.
Can I fix it to a rented wall?
It needs screws into the wall, so check your tenancy first. There is no adhesive-mount version.