1. Perhaps. It would be best to reduce the bureaucratic process required for new vaccines etc., but still ensure the quality and safety one way or another. Labyrinthine paperwork makes no one's job easier, not even the paper manufacturer's.
2. I would not trust the military to maintain order, except in hotbeds of public dissent and the total dissolution of order. Then, and only then, should the military be summoned. Ordinary law enforcement may be more trusted due to its inherent lack of capacity to deal with citizens in the same way the military does.
3. Bureaucracy and red tape might, depending on how heavily regulations are enforced. Under the inherent nature of the free market, if it is not tampered with by political and bureaucratic maneuvers, the wealthiest will receive first priority. They would almost certainly be able to pay for the vaccine first, but they will approach with caution. I'm willing to say they are willing to wait a little while to see if their middle-class counterparts have any adverse reaction to the vaccine. It is better to be safe than sorry.
4. Yes. However, while there may be Radicals and devout fanatics, the vast majority of the people are sensible enough to refrain from listening to opportunist populists. I'm sure people vote with their hearts and then their minds in calmer circumstances, but they will realize these upstart prophets can do the talking but not the walking.
5. Yes. We face a demographic crisis in the developed world because there's not enough space to live comfortably, not enough time or money to take care of a child and live well-off, or any real economic benefit from keeping them.
On the other hand, such a sudden and abrupt destruction of the human population might dramatically increase birth rates, because this frees up a lot of open (but developed, depending on where they are) space for the remainder. The remaining labor force would be in high demand, like the brief period of time after the Black Death where labor was highly valued due to an obvious scarcity. That means two problems with having children are solved: the money problem, and the space problem. Now, there's a time problem, but assuming that the demand for labor is high enough to stimulate a strong rise in wages, that means one parent can still look after the child while the other earns a living.
As a result, I see our civilization prospering and restarting. Our high-tech system would be forced to develop, such as industrial machinery required to supplement the scarce labor force (initially), or increased communication networks and means of travel, since it is likely smaller centers of population, like obscure villages with a lacking environment and no access to anything beyond basic resources, which would still be hard to use once population recovers, will be swept away, while larger towns and cities develop.
Surely, there would be a shortage of skilled labor, but assuming we keep good records of our predecessors, it would not be disastrous enough to cause us to behave like cavemen relative to our time.
The system which allowed such a collapse to happen would surely be blamed for its deficiencies, and they would be forced to establish a more orderly response system. Medicine and our bodies will have to adapt to this new disease, which is good. It all depends on how fast medicine and our humans can recover and develop.
