Is it permissible to install teeth in place of missing ones, knowing that this will require altering the spaces between the original teeth, and does this fall under the ruling of "those who make spaces between their teeth for beauty, altering Allah's creation"?

There is no religious impediment to installing artificial teeth in place of missing ones that have not erupted, because this falls under the rubric of treatment and removing a defect, which is permissible in Islam. It is not considered altering Allah's creation, which is forbidden.