The tours are not available at this time of the year, but is it still possible to hike up to Vikingsholm and walk around inside without doing the tour? As in just doing a self-guided tour.

Nope, you can walk around most of the outside but not the courtyard.

Absolutely not. This is a state park and the home is like a museum you are only aloud entry upon purchase of a guided tour.

No. You must do a tour to visit the interior. You can hike down to the grounds though without doing the tour.

Yes, but you may have to park on the street.

Yes, you do need to purchase a ticket in order to tour the home.