Do I need a mattress encasement or protector for this DIY mattress?

I bought 4" of latex and 8" of pocket coils and a natural wool/cotton cover from SleepEZ/LatexMattressFactory. The cover is only recommended to be spot cleaned and not washed. Does this mean I should get an encasement and protector and sheets rather than just sheets?

Yes, you’d still want to place some sort of a mattress pad or mattress protector over this, just like any other mattress, to help maintain a healthy sleep environment and minimize the body oil/skin cells/miscellaneous detritus that gets into your mattress. The mattress cover you purchased isn’t meant to be something to be frequently cleaned and to catch those materials. It’s primary role is to encase your componentry and help keep everything in place.

Is it necessary to have both the encasement, and the protector (5-sided), or should I bother with an encasement only if I find bed bugs? So far in 10 years, I’ve never had bed bugs and that seems to be the selling point of encasements.

A mattress protector that is offered as a zippered encasement can be helpful for the prevention of ingress/egress of dust mites/skin cells/bed bugs. Those would be the main reasons to purchase a zippered encasement mattress protector. I find them most helpful for people with severe dust mite allergy issues. Otherwise, most people are fine with the mattress protectors that fit on the same as a fitted sheet.