Can I get chicken pox from a child who just recovered from it a few days ago?

Yes. If the lesions are not completely healed and you have not had chicken pox or your immunity is weakened it is possible. Once the crusts are dried and have fallen off it is not contagious. There is an incubation period of about one week.
Hard to speculate. Most adults are immune, even when they don't recall having cpx. You don't define "recovered in the kid.They are contagious from the day or so before spots start until a week after they do, and the last one crusts. The incubation period is 10-21 days, so you may find out later, if you get it call your doc immediately.It can be deadly in adults.
Not likely. They are contagious 1-2 days before they start pox until the last scab crusts. Not contagious after.