I shall start with a brief answer to the numbered questions.
As far as I am aware, there is no such fustian rule in a dusty grammar. But the finer points of the order of adverbs is not an area which i have considered. I can imagine fine differences. Of the alternatives given in the question, I marginally prefer the first, with "regularly being shared". The second, "being regularly shared" might be construed to involve not just that regularly a lot of personal information gets shared, but that lots of personal information gets regularly shared, which is very slightly different. But I would not go to the stake on this. I probably would not have noticed, had the question not been raised. In context, it is abundantly clear what is meant.
The phrase fragment "...with personal information regularly being shared..." is (I think) an absolute phrase, and "being" (or "being shared") is a participle, qualifying the noun information. It is not, in my view, a gerund. I have already explained this in comment and a reply, and shall say more below.
I cannot comment on the diagramming question. I was educated in Britain, and this is a phenomenon of US school education, unknown in the UK. I first encountered it in reading Noam Chomsky's revolutionary book Syntactic Structures, in which he uses the idea of diagramming as a way to present his idea of what he called 'deep structure'. My understanding of grammar came from the study of Latin and ancient Greek.
There is a term, used in two sources I have found: that of the absolute phrase. Examples of this are found, for example, in English Grammar (https://www.englishgrammar.org/absolute-phrase/). It explains:
When a participle and the noun that comes before it together form an independent phrase, the structure is often called an absolute phrase.
I should say that is a fair description of what I understand by the idea. It goes on to gives examples like these:
The weather being fine, we went out for a picnic.
The visitors having left, we went to bed.
It is easy to see that the first of of the two, express a causal relationship between the fact that the wether was fine and the fact that we went out for a walk. It could have been expressed as:
Because the weather was fine, we went out for a walk.
The second of the two expresses a temporal relationship between the departure of the visitors and our going out for a walk. We did not go out because they had left. We just did so afterwards.
When the visitors had left, we went out for a walk.
In both sentences, the opening phrases involve participles, not gerunds. And there is a perfectly good reason for saying this. Take these two sentences.
Tired of watching TV, we decided to go out for a walk.
Here, as in the first example, the relationship is obviously causal. But here it is obvious that "tired" is a participle and that it qualifies the subject of "decided". It could not possibly be a gerund.
The second sentence can receive the same treatment.
Bidding farewell to our visitors, we went out for a walk.
This has the same temporal relationship, and it is obvious that "bidding" is a participle and qualifies the first person plural pronoun 'we'. It cannot possibly be a gerund. So why, in the two sentences in question, should it seem to be a gerund rather than a participle? Well, in both these sentences, the '-ing' word in question does not qualify any noun or noun phrase in the sentence! Oh, yes: and there is this wretched little preposition "with". With has to modify some noun in the sentence. But there is none that it can modify... except ** the whole phrase "with personal information being shared...". And that would make "being shared" a gerund, surely. A generation ago I should have been able to say that, if so, "personal information being shared" should be "personal information's** being shared. But the use of the possessive 's with the verbal noun has completed its process of dropping out of use.
The temptation to treat it as a gerund is the greater, because of the difficulty of the grammatically homeless phrase.
But there remains the alternative I mentioned at the outset: The "Absolute Phrase". What does this mean and where does it come from? I have no doubt that it owes its existence to the absolute phrases familiar to the many people who at school studied Latin and the slightly fewer who also studied ancient Greek. Latin uses participles in exactly this way, and it is called the ablative absolute. I am as sure as I can be that this accounts for the choice of the term absolute in this context. In Latin you might get a sentence like this:-
Obsidibus capris, Caesar exercitum in hibernias duxit.
Literally, "Hostages having been taken, Caesar led his troops into winter quarters."
This construction is used to express a temporal or causal connection. Ablative is the case used to express agency, cause or origin, and was explained in the grammar books with the words by, with or from. In ancient Greek, which had no ablative, the corresponding construction employed the genitive, which expressed not only possession, but also agency, origin and cause.
Where does this 'absolute' word come from? It is and always has been the word used for this particular Latin construction: the ablative absolute. Latin was the language of learning and scholarship throughout the middle ages and widely used in the 18th century: both Newton in England and Leibnitz in Germany wrote their seminal works in Latin. Even in the 19th century Latin was taught in private schools and grammar schools, and continued to plague such pupils well into the 1950s and early sixties. The following is a good explanation of it.
The ablatives of a participle and a noun (or pronoun) are used to form a substitute for a subordinate clause defining the circumstances or situation in which the action of the main verb occurs. The ablatives are only loosely connected grammatically to the remainder of the sentence, hence its name absolute (absolūtus = free or unconnected). Its connection with the sentence is, essentially, logical: syntactically, it stands alone.
There are, in fact, familiar phrases that work like this.
Weather permitting, we should make it to top.
That being said, I don't approve of what you are doing.
Given the presence of rioters in the street outside, they they decided to stay where they were.
All things considered, I think we had better leave at once.
But where did the with come from? My answer is speculative. The Latin ablative absolute was the bane of many a reluctant victim of the study of Latin. It was also the bane of teachers of Latin, trying to move their pupils away from the hideous literal translation of "obsidibus captis" as "with hostages having been taken". Well, out of the possible interpretations of the ablative, 'by' and 'from' seem unpromising, so perhaps 'with' will do. But since then the usage of 'with' introducing an absolute phrase has become increasingly common.
