Before answering the questions, I´ll say I am and incredibly weird christian very far away from what mainstream christianity think and I amprobably what people call syncretic.
1) With hundreds of thousands of demons, why are possessions so rare?
Demons are spirits that are unhealthy to humans. This entities favour anger, sorrow and other undesirable states of being so they can feed from them. They are generaly very dumb too.
Smart and powerful spirits use much subtle methods to influence material reality, which would be more effective because, well, they are smarter and more powerful spirits than the great great majority of weak dumb ones.
About the beliefs of MAINSTREAM christiniaty: No evil spirit can do ANYTHING without God´s consent. So possesions rarely happen.
If a possesion happens, God allowed it, and it could be for many reasons. The complete control of God over the evil forces is exemplified on Jesus, that could command demons and did so out of people.
One of the reasons why God permits possesions according to the church(catholic) is that it may be to remember us that the powers of evil are real and to make us turn more into him. In other words, God uses demons to make people turn to him to find love grace and what not so they can be happy and live eternally.
2) Do people often know that they are possessed? For instance how would I know if I was possessed?
Well, I personaly think that evil spirits influencve people very often, but it is an influence that is very mild in comparison to the dramatic displays in exorcism and such. I wouldn´t say you can now for certain, but if you in general feel loving towards everyone, rarely get angry and when so is not that strong, don´t hate, have a generaly peaceful state of mind, etc you are mostly likely untouchable to wicked spirits.
Now, I do believe exceptions can happen, when very powerful demonic forces want to corrupt a person that is generaly very good but has some strong form of dualism in him and this is how I view the saints that have struggled with the demon. Strong and smart demons are required for this kind of things, not the weak dumb ones.
3) Can Christians be possessed?
Absolutely! and the pwoer of Christ is around the best and most powerful exorcising tools around if you ask me. Sheer force of will and trying to pray to God done by the victim and/or friends helps a lot too IMHO.
3) 4) Would Satan allow his demons to possess his followers in a harmful way, or would such a thing be counterproductive to Satan's goal of leading people away from Christ?
It depends on how you define "satan".
For starters, I don´t believe there is one single spirit that has ever had that title. People that worship satan as an agent of freedom, knowledge, etc would in my opinion not be worshipping the same deity at all that christians call satan. Satan is from a christian perspective an EVIL spirit, that DOESNT HAVE the wellbeing of humanity in his will, but the OPPOSITE.
Now when talking of powerful smart evil spirits they will do whatever best suit their goals. Sometimes this will be possesion, sometimes not, the different ays, moments, manifestations and motives carried out by each possesion would vary widely.