First and foremost, all four of those sentences are in fact grammatically correct. What Google demonstrates is that a.2 and b.1 are really awkward and therefore much less common.
In terms of the implications of the words themselves:
Opportunity often implies that it is an offer being extend or provided to you by a specific entity or person who is providing you with means or privilege otherwise unavailable to you. Sometimes there is no other party offering the opportunity, but there is always the sense of urgency, that it is a limited time deal, something that must be seized or it will be lost. There is generally a positive connotation: an opportunity is a good thing, something to be excited about.
Possibility simple implies that it is something that can happen, either because you decide to make it happen or because circumstances cause it to happen. There is no implicit positive or negative connotation here. There is also no implicit implication of the likelihood of the event happening. If it is possible, it may or may not happen.
In general, opportunities imply possibility, but not the other way around. If you are given the opportunity to go to Norway, then of course it is possible. But if you lived within driving distance of Norway, you wouldn't usually refer to your possibility of driving there at any moment as an "opportunity".
Some examples:
There is a distinct possibility that I may lose my job next week, but I have been given the opportunity to work at an exclusive Swedish design firm, so I'm not too bummed.
I really needed to talk to my professor. He was a busy man, so this was nearly impossible, but I happened to see him walking alone one day, so I seized the opportunity.
Interestingly, this Ngram suggests that in fact the usage of "opportunity to go" and "opportunity of going" have reversed in the last 200 years, with both being approximately equal in use around AD 1890-1900.
"Sole" means "the only one". "Only" means the set is restricted to the identified members, but there could be more than one in that set.
That is, in all your examples, you could use either "sole" or "only" and the sentence would mean the same thing, EXCEPT for #5. You cannot say, "The two women were the sole survivors" because "sole" means only one. Effectively you're saying, "Only one person survived: Sally and Jane", which makes no sense.
That said, we commonly say "He was the sole survivor ..." or "the sole heir". But I've never heard someone say, "It was the sole useful book ..." or "I am the sole child ..." I'm hard pressed to think of a rule for when one is used versus the other in cases where either is valid.
Best Answer
All four sentences are grammatically incorrect if no way is dropped.
One can say:
In this case, the two phrases to prove and of proving seem to me to be fungible. They're just a style preference.
Have you seen any documents that say There is to prove that he was {right / wrong}? Please provide a link.