I have noticed that my way of analyzing English differs to an extent from what is taught in the Anglo-Saxon countries. This is how I see your examples:
There was never any stopping it. Stopping is a gerund. It is used (without a verb immediately before it) after a form of to be (was), and that is typical of gerunds. Stopping also has an object (it) in the sentence, which is quite common for a gerund.
He has the gall of a shoplifter returning an item for a refund. Returning is a present participle. He has the gall of a shoplifter is a complete main clause. By that I mean it has a subject (He), a finite verb / a main verb (has) and an object (the gall of a shoplifter). It qualifies as a sentence on its own; it is in no way incomplete. Returning an item for a refund is not a sentence or a clause because it has no finite verb. Returning connects it to the preceding clause and the meaning is thus made clear.
In traditional European grammar returning is said to begin a clause equivalent, but I know that that term is not used a lot in the UK and the USA. In your sentence we could have a subject and a finite verb instead of the participle: He has the gall of a shoplifter who returns an item for a refund. Participles are very often used to replace all manner of subordinate clauses. Some examples:
When turning a corner, I saw a lorry hit a car. (= When I was turning a corner, I saw a lorry hit a car.)
Although living in Spain for years, he didn't learn Spanish very well. (= Although he lived in Spain for years, he didn't learn Spanish very well.)
That's a good car compared with mine. (= That is a good car if it is compared with mine.) Compared is a past participle, not a present participle because the clause is in the passive voice.
He could never see a belt without hitting below it. Hitting is a gerund. This is perhaps the easiest of your sentences for a layman to analyze because without is a preposition and all prepositions must be followed by a gerund. Examples:
He had an opportunity of visiting his uncle. We succeeded in reaching our destination in time. I'm accustomed to getting up early.
Wrestling with words gave me my moments of greatest meaning. Wrestling is a gerund. It is the subject of the clause/sentence. Gave is the finite verb. A present participle cannot act as the subject of a clause. Similar examples:
Swimming is great fun! Reading detective stories was one of his hobbies. Writing letters isn't what I like.
Cheers, CB