Use whatever blows your skirt up.
The bottom line with oil is your guns don't need the surfacants, dispersants, pour-point depressants, over-base additives (to neutralize combustion acids), anti-oxidants (for the oil itself, at high temperature), and coiled-polymer viscosity modifiers that the vast majority of engine oils (synthetic ones included, for the most part) contain. Engine oil is engineered for temperatures your gun will never see, pressurized dynamic film lubrication your gun isn't capable of producing, flood lubrication that would't suit a firearm, a sump, and filtration. Engine oil is not designed to operate at 70 degrees, it is designed to operate at ~200 degrees. The only additives in engine oil that suit firearms are the anti-corrosion additives and EP additives which are becoming increasingly scarce as engines are produced now with fewer sliding surfaces and ever more additive-sensitive exhaust catalysts and sensors.
Straight non-detergent 30 wt conventional lawnmower oil is good. Straight, heavy, laxative mineral oil is also good. Lucas synthetic oil addtive is also good, because all it is is a group IIIA hydrocracked oil and some tackifier (polybutene) and none of the other stuff....which incidentally makes it a lousy engine oil additive because it dilutes the additive package in the balance of the oil used. I say the synthetic is what to use because it is far less sticky and viscous than the regular stuff.
Dexron III ATF is OK because it is a paraffin base and it does not build varnish. It's about an SAE 7-8 weight oil. As far as I know, there is no true "synthetic", group IV transmission fluid because all the syns are too slippery for wet clutches.
Grease. Well, that's a deep subject too. RIG grease is old but good, simple technology, akin to Alox and Cosmoline, and for the same reasons: It has a calcium soap thickener which is naturally very highly hydrophobic and it adheres well to metal. Any good marine grease, particularly boat trailer bearing grease, serves this purpose. So do many of the GP aircraft greases, even if they aren't calcium based, and you don't have to contend with calcium caking or the grease drying out.
For the most part (aside from thickeners) you have sticky grease and buttery grease and different thickness grades to choose from for your guns. Most guns don't like a sticky grease at all, so buttery is the way to go. Lithium or lithium complex is my choice, a #1 if I can get it or #2 (thicker, typical automotive and general industrial use grade) if that's what I have. In some rare instances I use a #3, sticky, EP, clay-based synthetic with organic moly EP compounds, but that is reserved for and applied sparingly to high-pressure bearing points that don't travel much like sear tips, hammer struts, and bolt lugs. Lubriplate MAG-1 is a #1 grease that is very suitable for guns. My M1A does not like #2 grease so I use a Mobillith SHC 220 which is NLGI #1.5, full-synthetic PAO base oil, and lithium thickened. It is buttery and adhesive but not sticky. My AR buffer springs also prefer a #1 or 1.5 grease as opposed to the usual #2. MAG-1 Lubriplate has a polymer additive similar to most electric motor bearing/bushing greases which is critical for ensuring an easy start from stop. Most automotive greases will sort of gel in place at rest and require some extra force to start the parts moving. At the end of the day hashing this out you'd be hard-pressed to beat the PAO clay-based Mobilgrease #28, or Aeroshell #7 (clay/ester) or Aeroshell #33 (lithium complex, PAO/ester blend), or the Mobillith SHC 220 lithium/PAO for your guns. Anything that is recommended for aircraft jackscrews is going to have all the properties, including not being too thick, that guns need. You won't have to worry about the stuff drying out, leaving the building, corroding, staining your clothes, or caking.
Hoppe's gun grease is crap. There's far better stuff than RIG. Mobil 1 bearing grease in a can is for disc brake wheel bearings, and today's not-really-synthetic version is not even very good for that. Most of the gillions of gun-specific, single-purpose products are pretty good. Engine oil, meh. Ed's Red as a cleaner, excellent (with reservations about adding lanolin, unless you live in hot, humid environments). Don't even get me started on how much I HATE CLP.