Regarding SAP - Tests after WW1 persuaded the British that "Semi-Armour Piercing" was the best choice. It was designed to penetrate less than 1" of armor plate and then explode instead of exploding at first impact. Their tests convinced them that British destroyers would engage enemy destroyers which had stronger plating or very light armor over critical spaces, such as engineering or magazines. The intent was to enable British destroyers to more effectively stop their enemies.
However, in actual combat, the SAP shell did not perform as expected. The problem is that even in calm seas at short range it proved really hard to hit those specific points of an enemy unit. For example, at the first battle of Narvick, five British "H-class" destroyers engaged 10 German destroyers. The range was short, the sea was calm and the Germans thought that the British shells were defective because unless they hit something substantial, they would go right through the German and explode beyond the target. The problem was that the German ships were built as lightly as possible in order to obtain speed and so the SAP didn't have very much to hit. Captain Donald MacIntyre's 1962 book "Narvik" discusses the problem at some length.
On the other hand, SAP was the perfect round for engaging a surfaced submarine. The shell would penetrate and explode inside the pressure hull - something an HE round wouldn't do. One hit from an SAP that sub's war was over.
Regarding "Separate Ammunition" - the cutoff between fixed and semi-fixed or separate ammo seems to have normally happened at about 4" (102mm) for the Allied powers during WW2.
For example, the British 4" (102mm) antiaircraft gun fired exclusively fixed ammo. See
Britain 4"/45 (10.2 cm) QF HA Marks XVI, XVII, XVIII and XXI However, they had anti-surface weapons of the same caliber that fired separate ammo. In fact, if you look at any Flower-class corvette, you see the same angle irons for holding shells. In fact, a picture showing them can be seen here -the last picture on the page.
Britain 4"/45 (10.2 cm) BL Marks IX and X Pictures
But the ubiquitous American 105mm howitzer round was considered "semi-fixed" even though you almost always see it handled like a "fixed" or single round. That's because the powder case wasn't crimped onto the shell. The case was separate so that before loading the round, powder bags could be removed from it in order to manipulate the shell's trajectory and range. Sometimes you see a shot of the gunners removing bags before putting the shell on the case. The round was then handled and loaded as a single piece.
M101 howitzer - Wikipedia, the free encyclopedia
In another example, except for submarine use, the widely used American 5" (127mm) was definitely "semi-fixed" in any version used since the powder case was totally separate from the shell - the rammer punched the shell and case in together. NOTE: The submarine 5"/25 lacked the rammer, and in any case was not intended to engage the enemy at any significant range. Accordingly a special fixed ammo was created for their use. But that's the exception.
I'm not an expert, but I think that cutoff for Axis power use was around 105mm, but the Germans in particular had larger caliber weapons that fired fixed ammo - a 128mm flak gun comes to mind.