On Locking Ranges using "$"
The first problem I see is the $ sign is placed before the row. This tells the spreadsheet that it needs to LOCK that row (2 & 12) from changing numbers. The drag-handle operation relies on gsheets to find a pattern in your selection and repeat or continue it. If it doesn't find a pattern it repeats the elements you have selected. I tried what you were doing but with the dollar sign locking the column instead and Gsheets does not skip every 10 rows so I believe the handle-drag-operation just won't work for what you are trying to accomplish.
The following only applies if there are exactly 10 rows difference between each reference ( I do NOT recommend this approach )
If you want to use the handle-drag operation try this: =INDIRECT("Sheet4!$A"&(ROW(A1)*10-10+2)) & in the cell directly beneath that one: =INDIRECT("Sheet4!$A"&(ROW(A2)*10-10+2)) Select both of those cells and drag downwards. It will give you the desired effect.
Keep only one set of Headers in Row-1 if you plan on doing Maths!
Although your spreadsheet is pleasing to look at it is not in a typical TABLE format. A better way to list your data is to have only one Header where the fields listed are: DATE, Booking-Time, Description Of Booking (leave blank if none), Choice 1, Choice 2, Choice 3, Price. Every Row after row one should be an instance of what your headers are describing ( in this case bookings ) where each field/Column defines the characteristics of that instance.
Here is a break down of the choices I made with your data:
The "Daily Totals" sheet should have 6 total columns/Fields: Date, Choice 1 Daily Sum, Choice 2 Daily Sum, Choice 3 daily Sum, Total Daily Sum, # of Bookings that day.
In "Daily Totals" Cell A1 you can have the Formula =UNIQUE(Source!A2:A); replace A2:A with where your date column/Field is in sheet "Source" In "Choice 1 Daily Sum" I used the formula =SUM(FILTER(Source!D:D,$A2=Source!$A:$A)) ; Where D:D is the Choice1 column in sheet "Source", $A2 is the date column in sheet "Daily Totals," and $A:$A is the Date column in "Source"
This formula can be copied down for each row corresponding to a new date, and right for each Column/Field ( Choice 2 Daily Sum, Choice 3 daily Sum, Total Daily Sum)
In "Daily Totals" Sheet for Column "# of Bookings that day" I used the formula =COUNTA(FILTER(Source!C:C,$A2=Source!$A:$A)) ; Where C:C is The "Description Of Booking (leave blank if none)" column in the "Source" sheet, $A2 is the Date in the "Daily Totals" Sheet, & $A:$A is the Date in the "Source" sheet.