"Can you use those Accushift levers with any Suntour freewheel or do you have to have an Accushift freewheel?"

My first indexed bike came about when I bought 7-speed Accushift road levers and derailer paired with an old Pro-Compe Ultra-6 freewheel.
The shifting was good across the range, never had to do any adjustment compromises. I used a UG Narrow chain, which is dimensionally the same as current HG70/91 models.

The only incompatible Accushift parts were from the 3040 Accushift Alpha group, a low-end MTB system.

Shifting performance on any old Suntour freewheel is greatly enhanced with a Shimano 9-speed chain, since Suntour's thick cogs and narrowed spacing between the largest cogs really need the extra clearance.

I've also paired the 7-speed Accushift levers with a pre-index Shimano Crane/Tourney-type derailer, which indexed perfectly on a standard 6-speed freewheel.