Is it permissible to marry a committed and well-mannered girl who, however, mixes with her non-mahram relatives and sees no impermissibility in that, and is there any objection to her leaving her phone number at the hospital?

It is permissible for a woman to speak with a non-mahram man out of necessity, provided that the extent of the necessity is observed and not exceeded, so as not to fall into corruption or open a door for Satan. Believers must comply with Allah's command and make their desires subservient to His law. If marriage is facilitated, the husband must prevent his wife from unlawful mixing, and she must obey her husband in this.