They are all adjuncts ('adverbials' in traditional grammar) in clause structure - the first two are depictive giving descriptive information, and the last one resultative expressing a subsequent situation (CaGEL p1224).
In the first two the situations are simultaneous and could be paraphrased crudely as:
He was lying on the ground and staring into the sky ...
Not learning French is my biggest regret.
*Not to learn French is my biggest regret.
Really, the negative is just a distraction. You can't say
*To learn French is my biggest regret
The reason is simply that the complex predicate adjective be one's biggest regret does not allow an infinitive subject complement. If if did, it might ...
At its most basic without a full lecture:
Gerunds are used when actions are real, fixed, or completed. "I enjoy cooking."
Infinitives are used when actions are unreal, abstract, or future: "He wants to swim."
...........but there are loads of exceptions!
If 'committed' always takes 'to' as a preposition, then it is always followed by the gerund (when it is followed by a verb form); prepositions are never followed by infinitives. (That's what makes 'to' tricky in English: it can be a preposition or part of an infinitive.) If you can look forward to a thing--a wedding, a party, the weekend, etc.--then 'to' ...
Taking bracketed adjunct and supplement out of the first sentence given
To acknowledge the equality of black folks
[so it was believed],
accept a lowered status for rural whites.
To acknowledge the equality of black folks would accept a lowered
status for rural whites.
simplify with demonstratives and take ...