Positive occult blood (rarely abbreviated as POB) is a medical term used to describe the presence of blood in the stool that is not visible to the naked eye. This means that the blood cannot be detected by simply looking at the stool, but can instead be found through medical testing. One common test used to detect occult blood is the fecal immunochemical test (FIT). During this test, a small sample of stool is collected and tested for the presence of blood using antibodies that react specifically with human hemoglobin. If blood is detected, it means that there may be bleeding somewhere in the digestive tract. There can be various reasons for the presence of occult blood in the stool.
To begin we need to ask whether the Pact of Blade weapon is a magic weapon without benefitting from the Improved Pact Magic invocation. This answer says that it is not and therefore is not destroyed by an anti-magic field. The test for determining if something is magical is as follows:
Given that I ve concluded above that a weapon being a hex weapon doesn t change anything regarding what detect magic would pick up on, then the magical weapon s magic would not be altered by Hex Warrior in any way. Pact weapons would probably show up as magical, but as for the school, if there is one, that would be up to the DM; hex weapons wouldn t, unless the weapon is magical already.
There can be various reasons for the presence of occult blood in the stool. It can be a sign of underlying conditions such as colorectal cancer, polyps, ulcers, diverticulosis, or hemorrhoids. In some cases, the presence of occult blood may be a false positive, caused by factors such as certain medications, diet, or gastrointestinal bleeding from the upper digestive tract.
Pact weapons would probably show up as magical, but as for the school, if there is one, that would be up to the DM; "hex" weapons wouldn't, unless the weapon is magical already
These warlock class features call out these weapons as magical. From Pact Boon - Pact of the Blade (PHB, p. 107):
This weapon counts as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.
Whether this means that it is actually magical for the purposes of detect magic, this isn't clear, but this gives a DM something to work with to make a ruling. I'd personally rule that the pact weapon is inherently magical, but that isn't explicitly stated in the text.
2. Magic weapon transformed to Pact of the Blade weapon (via the ritual), compared to before the ritual
As for your second point, the weapon that you transform into your pact weapon is explicitly called out as being magical before you do so (PHB, p. 108):
You can transform one magic weapon into your pact weapon by performing a special ritual while you hold the weapon.
Since it is already a magical item, it would show up as magical to detect magic, but it would have done before it became a pact weapon as well. Whether being a pact weapon has any impact on the school of magic it registers as would be up to the DM (see the last section of my answer for more on this).
3. Pact of the Blade's Improved Pact Weapon (eldritch invocation) and its created weapon with +1 to hit and damage
As for the Improved Pact Weapon invocation, we can turn to the general information on eldritch invocations (PHB, p. 107):
In your study of occult lore, you have unearthed eldritch invocations, fragments of forbidden knowledge that imbue you with an abiding magical ability.
That seems more clean cut to me; if an invocation is a magical ability, and the Improved Pact Weapon invocation affects the weapon, then the weapon is magical thanks to that invocation (at the very least, magical enough to show up as magical to detect magic, even if the magic being detected is the invocation, not the weapon).
However, this related question shows that not everyone views such as thing as so clean cut: Is a Warlock's Improved Pact Weapon considered a magic weapon?
4. Hexblade's chosen weapon, by touching a normal weapon after a long rest
The Hex Warrior feature of the Hexblade warlock says (XGtE, p. 55-56) doesn't mention magic at any point, so presumably the "magic" here is that you are more capable of using it that you "should" be, thanks to your eldritch magic, but the weapon itself appears to be no different. I would conclude that the weapon is nonmagical and you would get nothing from detect magic.
5. Hexblade's chosen weapon, by touching a magical weapon after a long rest, compared to before choosing it
Similar to the above, except that because the weapon in this case is already magical, it would show up as magical to detect magic, but no different than it would have done before the Hexblade got their hands on it. Given that I've concluded above that a weapon being a hex weapon doesn't change anything regarding what detect magic would pick up on, then the magical weapon's magic would not be altered by Hex Warrior in any way.
Regarding the School of Magic
At no point in any of the descriptions of these warlock class features is a school of magic mentioned, so if a DM were to wish to treat any of the above as magical for the purposes of detect magic, they would need to come up with a school of magic themselves, or conclude that there isn't one. For weapons that were already magical (as per #2 and #5), they might want to use that weapon for guidance, if it mentioned a school of magic, but to the best of my magic, most (all?) magic weapons don't typically do that, so once again, it's probably just up to the DM's judgement.
As for the Improved Pact Weapon invocation, we can turn to the general information on eldritch invocations (PHB, p. 107):
It is important to diagnose the cause of positive occult blood to ensure appropriate medical intervention if necessary. Additional tests, such as a colonoscopy or upper endoscopy, may be performed to further investigate the source of the bleeding. Overall, the presence of positive occult blood is something that should be taken seriously, and anyone experiencing this symptom should consult a healthcare professional for further evaluation and guidance..
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