BNC, in a nutshell:
First, while BNC connectors are primarily made for coaxial cable, they are not tied to any particular type or IMPEDANCE of coax. Thus there are BNC connectors that are made for RG-58, RG-59, various miniature (1/8") coaxial cables, and probably a bunch more. And the 50 and 75 Ohm varieties are certainly common, but many have been made for use with 75 Ohm cable intended for composit video (30 Hz to 5 MHz) without having any particular specified impedance.
As I more or less say above, they are made to PROPERLY fit only one or perhaps just a few types of cable. Thus a BNC connector that was made for RG-59 cable may not work well with RG-58.
Oh, and they come in both solder and crimp styles. And I have seen and used some that are a combination of the two with the center connector being soldered on and the shield crimped. Oh, and then some with the shield being trapped between two parts the screw together. There are many manufacturers of BNC connectors.
You asked about fittings. There are Ls, Ts, connectors, and pass throughs. Ls come with two males, two females, and one of each. Ts come with three males, three females, and two of one with one of the other. Connectors are usually called barrels and are used to splice two cables. They usually have two female ends. But I have seen connectors with two male ends. Pass throughs are for running a cable through a box or chassis and also can have two females or sometimes a male and a female. They come with two kinds of mounts: a flange with four screw holes and a threaded barrel with a nut. Some of the pass throughs are liquid or gas tight.
I worked with them for over 45 years and I am sure I have not seen every variation. You can literally put a fortune into these accessories so I would suggest that you only buy them as you know you need them. Perhaps just get one or two BNC Ts of the F-M-F variety and, in your case, 50 Ohms sounds like the best.
Scope inputs:
With very few exceptions, scopes have BNC input jacks and often use them for outputs too. The vast majority of scopes will have a high impedance input with a 1 MOhm impedance with a few pFs of capacitance in parallel with that being the standard. Almost all scope probes sold presently and for the past 50 years will work with those scope inputs.
There are a number of different types of scope probes, including 1X or straight in, 10X or with a factor of 10 of attenuation, temperature, current, RF detector, and many more. As I said above, virtually all of them are made to attach to the standard 1 MOhm inputs.
Most probes will have a ground connection brought out to the tip where a ground wire or clip can be attached. Also most probes will have a number of interchangeable options for connecting to the circuit: clips, points, hooks, etc. These usually attach with a friction fit.
A 1X probe is just a piece of wire or coax attached to a BNC connector. They do not attenuate the signal but they also have less bandwidth than an attenuated probe. For the most part you can not utilize the full rated bandwidth of a scope with a 1X probe (or an unattenuated coaxial connection).
10X probes are the most common. They reduce the signal by a factor of 10 so a 10 Volt signal will show up at the input amplifier of the scope as a 1 Volt signal. 10X and other attenuated probes will or at least should have a frequency compensation adjustment. This is usually a variable capacitor at the scope end of the probe which is adjusted with a small screwdriver. The scopes usually have a square wave output available to allow these probes to be adjusted to properly matched to the scope's input.
100X probes are available but somewhat rare. They will allow even higher frequency compensation but at the cost of lower gain. For most applications the 10X probe is the best compromise.
Some scopes have special BNC jacks that allow automatic switching of the Voltage range depending on if a 1X or 10X probe is attached. A flat, ring like contact surrounds the ground shell of the BNC jack and a contact in the 10X probe grounds it to tell the scope a 10X is in use. Different numeric scales are lit via switched lamps for the two ranges. This is a nice touch but only the higher end scopes and probes have this feature. And you can always use any probe, just doing the math in your head.
50 Ohm scope inputs:
Yes, they exist. But, IMHO, a scope with a dedicated 50 Ohm input is a very limited scope. Not only are the regular 1X and 10X probes not useful, but the internal load will have a limited power rating so you will need to be careful not to exceed it. Such a scope is used only for RF power and other measurements and can not be easily used for normal circuit work. A 50 Ohm load placed at most places in any but RF circuits will look a lot like a short and may produce instant smoke. I have only seen one such scope in over 55 years in electronics and did not have much use for it.
50 Ohm input workaround:
In the pouch on my scope I keep some BNC Ts with one female and two male ends. The female end attaches to the high impedance scope input. The coax which connects the signal that needs a termination is attached to a male end and a BNC female connector with a termination resistor to the other male end. This terminates the coax just outside of the scope and is good enough for most purposes. This way loads of any impedance or power rating can be used.
But here is the long form:
https://en.wikipedia.org/wiki/BNC_connectorI tried to keep it as short as possible.