Plenum rated cable is only necessary for a space, usually above a
ceiling or below a floor, that can serve as a receiving chamber for
air that has been heated or cooled to be distributed to inhabited
You should be fine with riser rated cable unless you're pulling it through an area that is designed as an air return or supply.
Securing Low Voltage Cable(s)
If you are pulling these cables in new construction, you would want to use low voltage staples to attach your cables every few feet to a stud or other framing member. Obviously in existing construction, unless you remove a bunch of drywall, this isn't feasible. It is acceptable to use the hole(s) in your studs, top plate, bottom plate, etc. to support the cable run. The NEC makes this exception even for electrical cable in remodel work:
(B) Unsupported Cables. Nonmetallic-sheathed cable shall be permitted
to be unsupported where the cable:
(1) Is ﬁshed between access points through concealed spaces in ﬁnished
buildings or structures and supporting is impracticable.
Fire Block Penatrations
If you are drilling through a designated fire block (horizontal piece of framing in a stud wall), yes you are supposed to fill the hole with fire proofing caulk or spray foam. This might also apply to the top/bottom plates on the walls, but I do not know this for sure. Hopefully someone else here can shed some light on this. If you have enough access to caulk/foam the points where you drill holes, you could go ahead and do it for good measure.
In general, I'm not sure that much of NEC Article 300 even applies to low voltage installations. I recommend doing some reading related to NEC Article 725 or 800 where you will find the applicable references back to other articles, namely Article 300.