Woman can too be regent. So either Alexander's mother or his wife can be regent.
Not in England. There is a history of the mother of the heir (Joan of Kent for Richard II, Catherine de Valois for Henry VI) being excluded from the regency. Only having custody of the child-king AFAIK. Granted, William IV and Victoria both introduced bills to have their spouses serve as regent in the event of a minor heir AIUI.
As far as him being an invalid is concerned, I don't think that this would necessarily be the case. Victoria's great-grandsons, Alexei Nikolaïevich, Gonzalo of Spain, Leopold of Battenberg, Waldemar and Heinrich of Prussia all led fairly active lives. Sure, the activity alternated with periods where a minor injury laid them up for longer than normal. But they weren't being pushed around in a wheelchair 24/7