You can update all values after changing the geometry of the areas by running a query or two.
This will take each point, find an area that contains it (if there are several, take the first one) and put the ID of that area into point data (tested on model tables, I assume Point and Zona are drawings in Manifold 8):
UPDATE [Point] SET [Zona] = COALESCE((
SELECT First([Zona].[ZonaID]) FROM [Zona]
WHERE Contains([Zona].[ID], [Point].[ID])
After that, this query will recompute CalcField for each point (tested on model tables again, I only included one more table, you have to join in the others similarly, I also assume the formula you gave used coefficients from each table, not IDs):
SELECT p.ID, p.Area_sqm, p.CalcField,
FROM ([Point] AS p
INNER JOIN [Zona] AS a ON p.Zona=a.ZonaID)
INNER JOIN [Namjena] AS n ON p.Namjena=n.NamjenaID
) SET CalcField = Area_sqm * KoefZona + Area_sqm * NamjenaKoef
So, the process is - open the MAP file with all drawings and tables in Manifold 8, change the geometry of the areas, launch two queries one after the other to recompute values in points.
You can run the queries in sequence in one click using a script.
Hope this helps (at least a bit).