Redline:
A lot of tool and instrument makers used "proprietary" threads. At Brooklyn Tech HS, I remember us kids being told by about this by machine shop teachers and mechanical drawing/machine design teachers. Some referred to these threads as "bastard threads", which got us kids to laughing, hearing teachers use a word which would have landed any of us on the carpet.
To answer your first question, as to determining the pitch/thread form of very fine threads, a manufacturer would have sets of hardened, ground and adjustable female thread gauges (similar to how a die closes down to cut a tighter thread), and hardened and ground male thread gauges to set the female gauges to. To inspect and check form and pitch, an optical comparator would be used. This is an instrument which uses a strong "collimated" light source to throw a beam of light around the object being inspected. The shadow cast by the object is then magnified using lenses with a long focal length, and the overall size of the comparator is kept within reasonable size by the use of internal mirrors. The enlarged shadow is projected onto a ground glass screen. The result is a black, crisp "silhouette" of the part being inspected, greatly enlarged. The part being inspected can be mounted on a "stage" with micrometer adjustments for position. The ground glass screen often has "hairlines" to set up coordinate axis. The part being inspected, set on the stage, can be positioned with the micrometers. A draftsman would make a greatly enlarged template of the portion of the part being inspected, drawing it on clear film with India Ink. This would be secured to the ground glass screen of the comparator and lined up with the "hairlines". The part being inspected would be moved with the micrometers on the stage until it just lined up with the template on the glass screen. By using the micrometers, the amount the part deviated from the design geometry could be determined. The magnification in a comparator is on the order of a toolmaker's microscope. The comparators had the screen at a convenient height, and had a black cloth hood like an oldtime studio photographer's camera. This let the person doing the inspection see the screen sharply with no reflection or wash-out from light in the room.
I am telling you methods that are probably outdated. I imagine nowadays, it is some sort of close circuit video camera with electronic magnification that is used to pick up the image of the part being inspected, and possibly this is projected onto a flat screen monitor, superimposed on a CAD image of the thread. My dentist, an oldtimer and man with a lot of respect for the machine trades (his father was a mechanic on the NYC subways in the car shops) and for engineering, has some interesting discussions with me about similarities in our professions. He has a micro camera he puts into my mouth and a flat screen monitor on the wall. He shows me in clear detail what he has found and what work needs to be done, and shows me the work as it progresses (I have good teeth, so little to show in the way of dental work, but we both like to play a bit). I had some arthritis in my left wrist that was really a problem, so saw the local orthopedic/hand surgeon. He shot it up with cortisone. He had a digital X Ray with a flat screen monitor. I watched him poke the needle into my wrist joint with a nice sharp image and considerable enlargement. I saw this same doc a couple of years later for another hit of cortisone in the same joint. As he pulled my chart, I saw a sketch I had drawn him two years earlier. He had broken a bolt off in a tapped hole on his tractor. I had drawn him the step-by-step of how to remove a broken bolt, with a perspective sketch of an "easy out". The doc and dentist and I get on famously, with a mutual respect. The result is I can see where something like an oldtime optical comparator is probably obsolete in modern industry. 40 + years ago, the optical comparators (sometimes called "shadowgraphs") were in common use.
A firm like Lufkin would have had a complete toolroom, fully equipped to make their own taps and specialty cutting tools. Or, a firm like Lufkin would have gone to a manufacturer of taps and dies and ordered "specials", taps and dies made to their non-standard or "bastard" threads.
Where you find a tap like it is a hard question. The first thing I would suggest, if you have a lathe that can cut a thread of 50 tpi, and handle small, fine work, is to try making a "dummy" male gauge. I'd get a piece of mild steel, turn it to the OD of the extension anvil's threads, and try cutting a 50 tpi thread. Do the shop math for half depth, use a well stoned tool, and inspect the tool against a center gauge carefully to verify correct thread form. I have a toolmaker's microscope, nothing fancy, no stage on it, but I put things like threading tools on a white paper, aim a fluorescent drafting table lamp down at the paper to get good light, and check with the thread gauge. My eyes are just not what they once were, so I need plenty of help for this sort of thing. I use HSS toolbits for most work, and grinding a 60 degree unified national form threading tool is something I've done many times. When a thread form tool for fine pitch or "close fit" threads is needed, I usually wind up stoning the flanks (sides) of the tool to get it the last little way to make the fit with the center gauge.
On the other hand, the outside possibility is that Lufkin did something totally off the wall and used a thread form other than the unified national form (60 degree) thread in common use in the USA. I kind of doubt it, though.
I'd make a dummy gauge, as I noted, if for no other purpose than verifying that the mike anvils are 50 threads/inch and unified national form threads. This gets you nowhere in terms of replacement parts, but it gives you information to use to track down a tap.
