Obviously we can't tell from your description how your work book is set out hence I've made some assumptions and believe that the following code, placed in a standard module and assigned to a button, may work for you:-
Dim lr As Long
Dim StartDate As Long: StartDate = Sheet2.[B1]
Dim EndDate As Long: EndDate = Sheet2.[E1]
Dim cNm As String: cNm = Sheet2.[G1]
Application.ScreenUpdating = False
lr = Sheet1.Range("A" & Rows.Count).End(xlUp).Row
Sheet1.Range("A1:I" & lr).AutoFilter 1, cNm
Sheet1.Range("A1:I" & lr).AutoFilter 2, ">=" & StartDate, xlAnd, "<=" & EndDate, 7
Sheet1.Range("A2", Sheet1.Range("I" & Sheet1.Rows.Count).End(xlUp)).Copy
Sheet2.Range("A" & Rows.Count).End(xlUp)(2).PasteSpecial xlValues
Application.CutCopyMode = False
Application.ScreenUpdating = True
I've prepared a little sample for you to peruse at the following link:-
In Sheet2 you will see a Start Date in cell B1, an End Date in cell E1 and a name box in cell G1 (which represents your drop down cell even though I haven't created a drop down list in this cell). For now, in the sample, type a name in G1, select start and end dates from those listed in Sheet1, then click on the "RUN" button to see it all work. Mix it up as you like to see if the code does the task for you.
Change the cell and range references to suit your actual work book and, if you choose to try it in your actual work book, be sure to test it all in a copy of your work book first.
I hope that this helps.