And for a 15 MHz square wave you kinda sorta need about 150 MHz analog bandwidth and about 1.5 GSa/s, otherwise even a very good signal will start to look deformed.
Your other points about the probes are correct, but those points are far too simplistic.
The scope bandwidth you need is completely unrelated to a square wave's frequency; it depends only on the risetime/falltime. The traditional rule of thumb is tr=0.35/BW. Simple thought: when looking at one transition, the scope neither knows nor cares when the next transition will come along.
Secondly, the relationship between samples/s and bandwidth is also too simplistic. Many scopes have an "ETS" mode where the sample rate is much less than the signals' bandwidth; many years ago I used a 25MS/s to observe sub-nanosecond risetimes. First google result for a contemporary scope: https://www.picotech.com/library/oscilloscopes/equivalent-time-sampling
OK, granted, I guess that's why it's called "rule of thumb" ;-)
Of course I assume a square wave has "reasonably" (for some definition of that) steep edges, needing about enough bandwidth for the 7th to 9th harmonic to show a reasonable (...) image of the signal. Since there is a relation between maximum possible rise/fall time and base frequency of a square wave the base frequency also sets the lowest allowable bandwidth limit.
The second sentence is wrong, pure and simple: the transition time is independent of the period. As for the first sentence...
Er, no. Let me guess. Your background is either theoretical or audio or software; you don't appear to have used a scope for looking at medium speed logic signals (medium speed = any logic family introduced since 1980)
Very few people look at square waves; the primary place you find them is in textbooks, especially associated with introductory Fourier analysis.
Most people look at logic signals. There the key factors are signal integrity and logic value. Signal integrity requires ensuring the transition time does not violate the specs in the logic families' data sheet (valid voltages at all times, and minimum slew rate, and minimum pulse widths).
It doesn't matter if your signal transitions once per microsecond or once per year: you still need to have suitably fast edge rates. The device I was using today (a 74LVC1G14) specifies t
r<=2.5ns.
It doesn't matter if your clock rate is 1MHz or 1mHz. If there is a runt pulse of a couple of nanoseconds then it can misclock or fail setup/hold times.
You would benefit from
understanding Bogotin's rules of thumb
http://www.edn.com/collections/4435129/3/Bogatin-s-Rules-of-Thumb especially rule 0.
As for ETS mode, hmm, I think that's largely a (historical) kludge, although if properly implemented it should work for suitable signals. IOW - usually not.
No. I explicitly referred to a contemporary scope.
Of course, if you have excessive amount of money then you can very fast sampling rates. But engineering is learning how to use your tools and - to repeat the old aphorism - do for $1 what any fool can do for $10.
You just can't observe a not strictly repetitive (over many periods of the timebase) signal using ETS. And if you already know that much about the signal there are only a few things left to find out about it by using a scope.
If only things were that simple.
Repetitive
or pseudo-repetitive signals are required. Often signals can be arranged to be repetitive. If not then you can still get important information from pseudo repetitive signals. For example, eye diagrams are also a current technique that won't disappear soon - especially with bit rates >1Gb/s.
For anything non-repetitive, glitchy, not-quite-known-in-advance there is no way around having enough bandwidth and sample rate to capture the actual signal - at first go.
It certainly helps, but eventually you will run into circumstances where that isn't sufficient; at that point you have to become more creative in how you use your tools. You might as well use the same techniques at other times.