Here's a different approach. The heart of it was created by turning on the Macro Recorder and filtering the columns per your specifications. Then there's a bit of code to copy the results. It will run faster than looping through each row and column:
Sub FilterAndCopy()
Dim LastRow As Long
Sheets("Sheet2").UsedRange.Offset(0).ClearContents
With Worksheets("Sheet1")
.Range("$A:$E").AutoFilter
.Range("$A:$E").AutoFilter field:=1, Criteria1:="#N/A"
.Range("$A:$E").AutoFilter field:=2, Criteria1:="=String1", Operator:=xlOr, Criteria2:="=string2"
.Range("$A:$E").AutoFilter field:=3, Criteria1:=">0"
.Range("$A:$E").AutoFilter field:=5, Criteria1:="Number"
LastRow = .Range("A" & .Rows.Count).End(xlUp).Row
.Range("A1:A" & LastRow).SpecialCells(xlCellTypeVisible).EntireRow.Copy _
Destination:=Sheets("Sheet2").Range("A1")
End With
End Sub
As a side note, your code has more loops and counter variables than necessary. You wouldn't need to loop through the columns, just through the rows. You'd then check the various cells of interest in that row, much like you did.