My theory is that while technically anyone can become a Guardian, Ghosts are very picky. This is due to the early days of their mission, when the need for quantity outweighed the desire for quality. Early Guardians, possessing their personalities, sentience, and free will, could have morals, relationships, and pasts that'd endanger them and their duties. The dead, though, have no such complications. Whoever they were before died with them, as well as their pasts. Dead Guardians could be shaped into what the Light needed much easier than living people.

Yes, they could become gaurdians, but only if the ghost deems them worthy. Ghosts aren't tied to only one person, and if their gaurdian dies and is unrevivable, then the ghost my find a new guardian. This is seen in the case of Shin Malphur, who inherited Wards ghost. Not all ghost think alike either, so the reasoning for why they chose their guardians is nearly impossible to pin down. It also noted that we aren't our ghosts first pick as seen with the easter egg in the Cayde's stash mission. If they found someone else before they came to us who met similar standards they were looking for in us, we might not have been revived. How ghosts choose their guardian is kind of like soul searching. They look for someone who best compliments their personality and way of thinking.

No. All Guardians were people alive during the Golden Age that had an affinity for the Light. This Light is essentially the core of your being. Light can not be imparted to you. It must already exist within you. This Light is what Ghosts look for when finding a Guardian. Apparently each Ghost is tasked with seeking out a specific individual. Once they find you they grab hold of your dormant Light and use it to resurrect you. It's kind of unclear if you have to be dead to become a Guardian or some Ghost just took too long looking for you, but most of the references to the "birth" of Guardians suggest that they all had a "first death". Basically who will become a Guardian was determined a long time ago. It's not something a Ghost decides themselves. The only factor is how long it takes your Ghost to find you.

In theory yes, but the individual themselves have no control over it. Each ghost appears to be born with some pre-ordained idea of who their guardian must be, though they themselves don't know who or why. They search until they have their 'eureka' moment and then they turn the individual into a guardian by infusing that person's very being with light. Among others, Osiris has speculated that there may be some kind of unifying trait or traits that makes an one person ideal for a guardian, but so far all proposed ones appear incomplete or incorrect.