Issues
1.) You placed your "Probation" condition outside an IF statement does generating the error of "Wrong number of arguments..."
2.) AND formula cannot be used inside an ARRAYFORMULA because it will end the whole array together which will generate unexpected result.
Solution
1.) You can place your "Probation" condition before the "Confirmed"&DATE condition right after the IF(LEN(F3:F)..
2.) Instead of using AND. You can multiply both conditions which will return 1 if both conditions are TRUE.
You can use the formula below. This is working as expected on my end:
=ArrayFormula(if(len(G3:G),ROUND(IF(LEN(F3:F),IF(G3:G="Probation",0,IF(F3:F<DATE(2020,6,30)*(G3:G="Confirmed"),13,(13 - MONTH(DATEVALUE(TEXT(F3:F,"mmm")&" 1"))) * 13/12))),"")))
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