I don't have any information on the rise in popularity of acronyms and initialisms in the 1930s, though I wouldn't be surprised if the popularity of such expressions in the USSR and the rest of the Communist world (NEP, OGPU, CPUSA) had some influence on persuading people that such shortenings were cool, space-efficient, and modern.
In any case my answer deals with the other two questions posed above.
Early matches for 'V.I.P.'
I couldn't find anything earlier than Compton Mackenzie's Water on the Brain (1933). Here is the Mackenzie quotation in full context [combined snippets]:
"...I'll show you the way to Major Hunter-Hunt's room. The Chief will see you presently. At the moment," she threw a quick apprehensive glance back over her shoulder, and her voice sank to a tense whisper, "at the moment he has a V.I.P. with him."
Blenkinsop trying to solve the problem of these initials could only think of Volunteer Indian Police, Veterinary Inspector of...but Pigs sounded improbable, and he tried Volunteer Intelligence Patrol before finally, giving it up.
Miss Glidden seemed to divine his perplexity, for just as they reached the threshold of Major Hunter-Hunt's room she turned round and whispered through a pursed up mouth, ‘Very Important Personage,’ her eyes flashing with the excitement of inside information.
Multiple sources confirm that this book was published in 1n 1933, although the version I consulted was a 1954 edition. Of related interest, according to Eric Partridge, A Dictionary of Slang and Unconventional English, eighth edition (1984), Water on the Brain also contains one of the earliest instances of another initialism, h.s.:
h.s. 'Hot stuff', esp. in the sexual sense: from ca. 1930. Compton Mackenzie, Water on the Brain, 1933, 'She's h.s. all right.'
You can see that usage, in context, here.
The earliest match I could find for V.I.P. is from H.T.W. Bousfield, "The Only Girl Who Wasn't a Bore," in The Windsor Magazine, volume 79 (December 1933–May 1934), page 332 [combined snippets], which uses the abbreviation many times and, like Mackenzie, reports that the P. stands for "Personage." Here is the beginning of Bousfield's story:
There are few thrones left in Europe, and the fewer there are the more important they become to the rest of the world. Because everybody knows it takes a republican country to appreciate a prince. Therefore when a Very Important Personage visited India he was much more awe-inspiring than he would have been if Europe were still scintillating with crowns.
Something of the kind was explained to the Very Important Personage, whilst he had his bath, at Government House on the evening of his arrival.
"You can't surprise me," said the V.I.P., splashing noisily. "The Problems of Empire are too vast to be appraised in their proper perspective save by a—— You'll find the rest in my speech tomorrow."
Both the 1933 usage by Compton Mackenzie and the 1934 usage by H.T.W. Bousfield involve VIPs who are high-ranking British officials, but whether they are in the military or the foreign service or some other branch of officialdom is by no means clear—and my impression is that the authors don't think it matters much. The min point to be noticed here is that the shortening to V.I.P. does seem to have originated in British English and probably spread to U.S. speakers and writers during the war years.
'V.I.P.' applied to nonmilitary personages
During the war years, V.I.P. got a lot of play in the military, which may have inclined people (at least briefly) to suppose that it referred explicitly and exclusively to high-ranking military personnel. But Louis Shores, Highways in the Sky: The Story of the AACS (1947) [combined snippets]observes that during the war civilian statesmen could also be VIPs:
During the first month of Farman's mighty Goose Bay station's life, a series of visits by VIP's occurred. Now, a VIP is the military abbreviation for a very important person. Any one with the rank of Brigadier General or above is a VIP. So are civilian statesmen, notable in all walks of life. The first of these VIP's, although not complying with the technical definition given, was still a very important person to AACS. He was no less than Conas—Lloyd H. Watnee, no more than a lieutenant colonel, in spite of his world-wide responsibility. Watnee was followed by General Arnold, who inspected the station first-hand. Then came diplomat Molotov, of Russia, and Mayor LaGuardia, of New York. All came to look at this new highway center in the sky, and to praise the master building that was bringing airways Z.I. ever closer to Airways E.T.O.— the European Theatre of Operations.
In the same year, Collie Knox, It Had to Be Me (1947) [combined snippets] seems to use V.I.P. to refer to anyone important enough for Collie Knox to interview:
Budding newspaper-men are informed that there is a special technique in the art of interviewing, particularly if the victim is a V.I.P. (Very Important Person). I suppose there is such a technique, But it cannot be learned and conned by rote.
... If I myself have a special technique when interviewing a V.I.P., it is this — I never carry a pencil with me. After the V.I.P. has been talking for some time, I become aware that he is chafing visibly at my apparent idleness. I mean, I am not writing anything down. That piques the V.I.P. a lot. He does not realize that, all the time, I am committing some, though not all, of his statements to memory.
And again in 1947, we find an early instance in which someone with an agenda describes non-VIPs as VIPS. From Telephony (1947) [combined snippets]:
Here's the letter signed by Manager Ken Johnson, the plant man and operators, that brought the record turnout:
"Dear V.I.P. "This is going to be plain old-fashioned flattery—but we think you're a V.I.P. (V.I.P. was used by the Navy during the war for visiting Congressmen, Presidents, Kings, and other Big Shots. It means "Very Important Person.")
Now as a telephone subscriber and user, you're a V.I.P. to us and we're hoping that our service, operators and repairmen are V.I.P.'s to you. ...
Once telephone subscribers have become VIPs for purposes of a phone company PR campaign, it's hard to see why anyone else (including show-business personalities) would be exempt. But I daresay that Edward G. Robinson and Bob Hope went on USO tours to Europe during World War II to bolster troop morale, they traveled as VIPs.