Am not clear on some things..if you were under age drinking I am assuming you were 18 or over which is why a background check would show this - were you ticketed or arrested? I live near a LARGE university and under age drinking is common..some are given tickets - which don't show on backgrounds, but some are arrested which mostly depends on their behavior (acting crazy, trying to fight officers, etc). Also, you don't state what you mean by "reckless driving" - that is a board term..sounds like you were arrested for this? If ticketed it wouldn't show on a background..so, if I were a hiring manager for hospice or home health or similar type position where driving for the company/on company time was primary I would not hire you.
As far as a hospital..I really have to know the details..it sounds like, not to be mean, but you have a history and tendency toward getting into trouble. Either by immaturity of not thinking or immaturity of age or both so I would be hesitant to hire but I would look at the totality of your background. As far as the Red Cross..did you mean you are going to volunteer with them? I know factually they also run background checks and ask for driving records - at least the chapter where I live - so keep that in mind.
I'm also not clear on whose fault it was that 9 months passed - because frankly, anything to do to close this/keep up with it is really your responsibility since its your career and future that depends on things getting straightened out.
Also, it sounds like these issues are board reportable - are you sure your license is ok?
Sounds like you have a fair amt to work out - I'd make sure the background and license issues are taken care of before moving forth with job interviews
. Good luck.