Well, I'll try to answer your questions.
1. As background. The first three BWI-built units went to the Pacific with six 20-mm guns (two before the bridge and four amidships on main deck) for USS NICHOLAS and USS O'BANNON, and seven guns (same plus the elevated platform before the bridge) on USS CHEVALIER. During the Guadalcanal campaign EVERY SHIP wanted more AA-guns. Even army 50-cal MGs were installed on some ships ... and never cataloged. The 20-mm guns could be installed by tenders and even crews themselves. The Pacific Fleet units in theater (or elsewhere?) figured out that they could add one (or in at least one case two) 20-mm guns atop the pilothouse. So all three BIW units were so equipped. This really helped with forward fire arc. In the Fall of 1942, a directive was sent out authorizing an extra 20-mm for any destroyer with six or less 20-mm guns. Soon it appears many destroyers took advantage of the availability of extra 20-mm guns, and even more guns were installed. USS NICHOLAS and USS O'BANNON were authorized TEN 20-mm guns and USS CHEVALIER was authorized ELEVEN 20-mm guns, with three installed on the fantail. Dave McComb told me the story that the Gunnery Officer on USS NICHOLAS told him that they at one point had thirteen 20-mm guns. I don't always believe such claims when there are no photos, but since he was the Gunnery Officer, I thought he was more creditable. I looked and looked for photographic evidence or mention in like Deck Logs/War Diaries and found nothing. Plus I could never figure out WHERE they could have installed three more 20-mm guns on USS NICHOLAS beyond the ten already there. There just wasn't any good real estate left for them. Anyway, the Gunnery Officer had the "extra" 20-mm guns removed after he was the one who got wounded by a round going into the bridge.
2. The FLETCHER class when ordered, had four 50-cal MGs authorized along with the quad 1.1-in mount. All four 50-cal MGs were to be installed forward of the bridge, two per side. However, before the first units were anywhere close to being completed, the 50-cal MGs were deleted from the design and six 20-mm guns added. As far as I can tell, the FLETCHER's had the close to the same depth charge battery as completed from contract design. The only difference was they started out originally with only four K-Guns and fewer depth charges. The as completed the all the FLETCHERS had six K-guns and stated with that number. The total number of charges carried was increased in late 1942, but not the number of drop tracks and K-Guns.
3 and 4. The FLETCHER class DDE conversions that retained the Mk 14 torpedo tubes mount, only did so because the planned four fixed tubes and ASW torpedos for them were not available. The Mk 14 mount served to keep the planned displacement as well. That was the same reason that the DDE design retained the K-guns. Yes some, but not all, the DDE's had the Mk 2 TLS (Torpedo Launch System) or the PMGL (No kidding ... Poor Man Gravity Launcher on the fantail) for the Mk 32 torpedoes installed during the 1950s. See attached image of USS WALKER (DDE-517) in July 1956. Depending on IF they actually had/carried ASW torpedoes for their fixed tubes. When the Mk 2 TLS, with two Mk 4 "over the side" launchers, were installed, they replaced the six K-Guns in the same location behind the spray bulwarks making seeing them difficult. The Mk 2 TLS also, rather unsuccessfully, was used to launch the Mk 45 ASW torpedoes, before the Mk 32 ASW torpedo tubes were available.
5. The first batch of FLETCHER class DDE's that had the Mk 15 trainable HH installed (the ones with pole foremast), slowly had the Mk 15 replaced with the Mk 108 rocket launcher during refits in the 1950s' USS O'BANNON was the last unit needing the upgrade. The next to the last FLETCHER DDE to get the Mk 108, did so in 1957. The USN decided to wait and to upgrade USS O'BANNON during her FRAM II upgrade, which as is known, was cancelled. Expecting that she would soon be decommissioned, they didn't "upgrade" her to the Mk 108. The Vietnam War saved her for duty as a gunfire support unit along with most of the remaining all-gun FLETCHERs left at that time.
6. USS HAZELWOOD was in the Boston NSY from 1 August to 4 November 1957, and had the SPS-28 radar installed. In late 1958 she was selected to be the DASH test ship. She went back into Boston NSY and was so modified between 27 March to 12 June 1959. Photos of her near the end of her conversion are dated June 1959.