Two Cell Consideration for result

So when it says A/B then you want it to return both dates?

This goes in D2...sorry, not very pretty...somebody may have a cleaner formula, but this at least works:

Formula:

=IF(C2="A",TEXT(INDEX(Sheet2!$B$1:$F$14,MATCH(Sheet1!$B2,Sheet2!$B$1:$B$14,0),3),"d-mmm-yy"),IF(C2="B",TEXT(INDEX(Sheet2!$B$1:$F$14,MATCH(Sheet1!$B2,Sheet2!$B$1:$B$14,0),5),"d-mmm-yy"),IF(C2="A/B",TEXT(INDEX(Sheet2!$B$1:$F$14,MATCH(Sheet1!$B2,Sheet2!$B$1:$B$14,0),3),"d-mmm-yy")&" & "&TEXT(INDEX(Sheet2!$B$1:$F$14,MATCH(Sheet1!$B2,Sheet2!$B$1:$B$14,0),5),"d-mmm-yy"),"")))

Edit: Fat-fingered a keystroke...formula corrected.

Another method...
=IFERROR(INDEX(Sheet2!$A$2:$F$14,MATCH(Sheet1!$A2,Sheet2!$A$2:$A$14,0),MATCH("end "&Sheet1!$C2,Sheet2!$A$1:$F$1,0)),
TEXT(INDEX(Sheet2!$A$2:$F$14,MATCH(Sheet1!$A2,Sheet2!$A$2:$A$14,0),MATCH("end a",Sheet2!$A$1:$F$1,0)),"mm/dd/yyyy")&" to "&
TEXT(INDEX(Sheet2!$A$2:$F$14,MATCH(Sheet1!$A2,Sheet2!$A$2:$A$14,0),MATCH("end b",Sheet2!$A$1:$F$1,0)),"mm/dd/yyyy"))