Set your conditional formatting by formula. The formulae I think you want to use, based on the provided workbook, are as follows.
1. =(IF(AND(E5>(TODAY()+15),E5<(TODAY()+30)),TRUE,FALSE))
2. =(IF(AND(E5<(TODAY()+15),D5="Open"),TRUE,FALSE))
3. =(IF(AND(E5<(TODAY()+7),D5="RR(s) Raised"),TRUE,FALSE))
4. =(IF(AND(E5<(TODAY()+1),D5="RR(s) Raised"),TRUE,FALSE))
Just make sure you use relative references (E5 instead of $E$5) and you should be good to go.
Cheers.
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