Correct. The group represents one, not its plural constiuent parts/members, though it is possible to say both visit or visits, but DL seems to disagree.
"Group" is a collective noun. It is always used with a singular verb in AmE while both singular and plural verbs are used in BrE.
Not strictly true, Even Americans say "the team play well together".
Yes, you have such a sharp ear =) actually, we normally link the three words and it will sound like /o grupu visitospital/ this o a slightly doubled /visito-ospital/, but still linked and difficult to tell it apart if one is not used to that yet.
You would simply says "The group goes to the hospital" otherwise it seems unnatural
There's a difference in meaning between "the group goes..." and "the group visits...". Goes means the movement towards, visits implies that there is a purpose of discovery or inspection.
(In US English "to visit with" can also mean having a long conversation with someone you haven't seen for a while.)