My first time replacing an AC/furnace filter:
The filter has cardboard around all its edges. That’s fine; so did the dirty old one I just took out. But this is more like cardboard that encases part of the filter screen itself, crisscrossing the screen in a lattice-like way.
Is the filter supposed to be inserted into the vent with this this cardboard encasing still on?
Yes. The cardboard lattice helps hold the thing together.
There should also be an arrow on the edge somewhere that indicates the direction of air flow. Since the filter is on the intake vent, the arrow should point to the furnace. At a minimum, filters should be replaced every six months, more often if you are in a dusty environment or have allergies. I use a Filtrete 2200 MPR filter that I order by the box from Amazon. I usually replace them every 3-4 months.
it is a stiffener/restraint.
some have wire or plastic to do the same function.