When I sort by field, empty fields are first
You can add a calculation field with a number result type such as: IsEmpty ( FieldthatcanbeEmptyHere ) and clear the "do not evaluate if all referenced fields are empty" check box. This returns a 1 if the field is empty and a 0 if it is not.
Then include this field in your sort order, sorting it in ascending order before sorting on the "FieldthatcanbeEmpty".
I also want to make a report which groups persons with anniversaries (50th, 25th, etc.).
You can sort to group records where the records in a given group have a common value such as a specific anniversary. This is often done for a summary report: Creating Filemaker Pro summary reports--Tutorial
is there any possibility to make a field with date with only day and month described?
Date fields have to have a day, month and year. But the field does not have to display all three values. You can select the date field while in layout mode and use data formatting options found at the bottom of the Inspector's data tab to specify a custom date format that only shows the day and month for your date field.