The Many Pronunciations of 法 Regarding the various pronunciations of 法字, how much of the below is accurate?
法 as in 法国:

Mainland: Fǎ
Taiwan: MoeDict has Fǎ, with Fà listed only as “旧音”, but my understanding is that most Taiwanese people use “Fà” even in modern times. Is this correct?

法 as in 法律 etc.

Always fǎ, as fǎr as I know.

法 as in 没法儿:

Every dictionary I have lists this as fǎr, but I'm pretty sure I've always heard it as fār (at least in northern China). Is this right?

法 as in 法子:

I haven't really heard this word used much, but listed here for completeness. MoeDict has this as fá, whereas mainland dictionaries list it as fǎ.
In all your examples, it is more productive to think of 法 as having the pronunciation of fǎ and that alone. But your examples illustrated 2 systematic situations where tones are customarily shifted:
Situation 1

when followed by a character with the third tone, third tone is shifted to second tone

In 法子, outside of colloquial use or regional dialect, 子 has the third tone. Thus 法 which usually has the third tone now has second tone. In actuality, you are welcome to pronounce both in third tone. But I suspect it will sound like second + third nevertheless.
Situation 2

In dialects

法儿 is a dialectic idiom for &quot;method&quot;. This Baidu page on Beijing dialect contains the word. To someone not from the area, 法儿 is still a legible word but the dialectic pronunciation is not known. Hence, the word may be used but pronunciation is the standard fǎ. The interesting thing is that standard Mandarin is supposed to have modeled after Beijing dialect. At least pronunciation should have. But that's clearly not being the case.
法子 (with 子 pronounced as the shortened neutral tone) is also a colloquial idiom for &quot;method&quot;. I would not be surprised if it is pronounced differently in different regions. I can see both second and third tone being used. (It can potentially mean a person in ~Warring Era due to 子. In that case it would not have this pronunciation of 子.)
Situation 3

When 1st tone is at the end of a word, in certain cases, it is shortened.

That's story of another day.


most Taiwanese people use “Fà” even in modern times. Is this correct?

Not sure about most. But I have heard it as Fà many times and have not heard otherwise from Taiwanese.

The people who study phonetics do have particular theories on why and when tones shift. They have more to do with way sounds are made -- not mechanically second -&gt; third. Phonetic studies in Chinese start way way way back as well, without which we would not know how to pronounce classical Chinese and classical poetry. But at this moment in history, tones are much simplified in Chinese. This double third tone -&gt; second + third scenario is one of the few 'rules' that are still prevalent.