Before we start, note that any pathogen that does all of these things is a stretch - it is nearly impossible for this to evolve naturally. Consider GMO bacteria or something. Therefore, my answer has inherent flaws - no matter what, all of this is somewhat optimistic, and somewhat unrealistic - but it's worth a try.
0) Hundred Year Statis, Rare Chance of Mutagens
These are tough ones.
No existing virus, bacteria, or other pathogen intentionally keeps its host alive - they generally kill them in the quest for reproduction or nutrients. This is trivial at best, but if your virus wants to avoid detection, it may nourish the body by rerouting nutrients through the bloodstream to the surface, all the while attacking less visible, internal organs. In this way, the vampires will stay young, while the disease can still eventually kill. The vampires, as opposed to infected humans, may survive if they have a particularly strong immune system - allowing them to pass the deadly phase and develop into monsters. The sleep for hundreds of years part could be attributed to slower metabolisms similar to hibernation, and the strong immune systems could stop other diseases from killing them.
This is much less likely - mutations usually occur in patches of cells - tumors - or in the womb before birth. Alternatively consider the immune system thing in the above passage - very few people survive (high mortality rate) but if they do, the virus stays with them.
1) Temporary Blood Rage
Option A: Gut Bacteria
The bacteria in your gut, according to a surprising number of studies, have an impact on cognitive function, behavior, and mood. One study even found a link between specific bacteria and aggression. Therefore, you could explain bursts of rage when too many bacteria enter the gut - changing mood and releasing adrenaline. These bursts would end if the bacteria eventually died or exited the gut. The problem with this explanation is that the "rage" would take some time to build - the connection is not instant - so you would expect a 30 minute gradual anger period followed by a 30 minute gradual cooldown.
Option B: Controlled Rabies
When humans have rabies - a disease that causes constant agitation that could suffice for a "blood rage" - they may have bursts of altered activity - that is, they are rabid sometimes - but much of the time they are fine. You can extend the idea of "bursts of altered agitation" to "occasional blood rages". While this may not be controllable by the vampire themself, it does produce the desired effect, and rather quickly.
2) Independent from Vampire, Used Against Enemies
Independence from Vampires: All diseases have stages
Most if not all diseases occur in stages - different symptoms occur or go away over time. Therefore, you could say that humans who die were too weak for the earlier stages, while those infected who became vampires were strong enough to survive to new stages. Voila, the same disease used in two ways - one of which is independent from vampirism altogether.
Use Against Enemies: Flesh Eating Tendencies
Another stretch, but worth a try. The most effective way for the Count to spread disease to his enemies is through airborne water droplets - basically cough or sneeze - although physical contact with the skin or stabbing may also work.
The problem with weaponizing diseases is that they will almost always take hours to go into effect. One solution is that the vampire virus has an early stage (which the Count avoided the worst of because of his strong immune system) similar to that of Necrotizing Fasciitis: the fast, flesh-eating bacteria!
Simply put, if the count touches one of his victims, their flesh may begin to tingle, then burn, then slowly eat away.
3) No Effects from Garlic or Sunlight; Nocturnal; Heart Stake
Garlic and Sunlight: Don't hurt people anyways
The idea that they hurt vampires is just a myth; no need to consider this for your plague.
Nocturnal: The eyes change, providing night vision and making sunlight hurt to look at
Some bacteria have been seen to stimulate cell growth, while rods are the cells responsible for the amount of "brightness" or "darkness" we see. After some handwaving, the solution: Your disease grows more rods, making sunlight too bright, but night vision a possibility.
Heart Stake: Causes tissue death
When the heart can't pump blood, the body's tissues can't receive oxygen. This will cause tissue death faster than the virus can repair, and kill the Count.
4) Heal Only if Not Under Attack, During Blood Rage, by Feeding
While Not Under Attack: Healing needs precision
While under attack, the vamp will be moving - it is likely that the process in which the pathogen repairs tissues is slow and precise, perhaps more so than normal repair because it's being conducted by something other than the body. Therefore, movement during an attack could prevent the precision needed to heal.
During Blood Rage: Hormones influence healing
Let's go with the "bursts of rabies" model of blood rage, because the gut bacteria one doesn't help in these circumstances. Cortisol, the hormone linked to stress and anger, is less present throughout the body as anger increases. You could handwave very slightly and say that cortisol, while keeping anger and stress managed, also helps keep the pathogen at bay - for whatever reason, it soothes the bacteria. Therefore, when you are angrier (blood rage) the pathogen is more active, and heals tissue faster / more actively. This model also successfully explains tissue regeneration during stasis: blood rage still occurs during stasis, at random intervals, allowing healing.
While Feeding: Ties into "healing needs precision"
Feeding requires lack of movement; you must sit still to eat. Therefore, feeding as opposed to walking will heal the Count the best.
5) Paleness; Boosted Physique; Changed Mandible; Changed Hormones; Violence; Resists Action Against Pathogen
Paleness: Develops over time as a result of nocturnal tendencies
Boosted Physique: Pathogen overcompensates when healing the host; muscles are built bigger, better, faster, stronger etc.
Changed Mandible: Modified Toxoplasma Gondii or Ophiocordyceps Unilateralis
The parasites listed above control the brain functions of their hosts to act in specific ways - to not be afraid of predators, to climb trees, etc - so you may be able to stretch this by saying your vampire bacteria tell the parts of the brain responsible for body repair to add a little bone to the mandible.
Changed Hormones: See above
Violence, Resists Help: See "Blood Rage" above
6) Morph Into a Batlike Creature and Fly Miles Away
This is not my area of expertise, but your best bet for morphing is having new tissue created by the bacteria as an approximation of wings, which are concealed under their clothing when not in use. See this question and answer for details about the wings themselves.
7) Bite Doesn't Mean Vampirism
Explained above: Onset of disease is in stages; only strongly immune individuals reach vampirism stages while all others die.
8) Pointed Ears, Fangs, Increased Muscle Mass, Superhuman Speed, Claws
Explained somewhat above: in addition to healing, the pathogen overcompensates, producing increased muscle mass, superhuman speed, and sharper nails. New constructions (ears, fangs) can be justified the way I talked about the mandible in requirement 5.
9. Don't Consider Blood Amount
Is saying "I'm not considering this" considering it, and therefore a paradox?