We have a Google sheet that we want a set number of rows and columns to be referenced into another sheet. However, those rows and columns are in the middle of the original sheet, and we want additional entries to populate in the new sheet without any outside cells from the selected cells in the original to populate in the new sheet.
You are probably looking for IMPORTRANGE
From the ImportRange functions page
Imports a range of cells from a specified spreadsheet.
spreadsheet_key - The long sequence of characters in the "key=" attribute of the URL. In the new Google Sheets, use the entire URL.
The value for spreadsheet_key must either be enclosed in quotation marks or be a reference to a cell containing the appropriate text. range_string - A string, of the format "[sheet_name!]range" (e.g. "Sheet1!A2:B6" or "A2:B6") specifying the range to import.
The sheet_name component of range_string is optional; by default IMPORTRANGE will import from the given range of the first sheet.
The value for range_string must either be enclosed in quotation marks or be a reference to a cell containing the appropriate text.
Notes A maximum of 50 IMPORTRANGE calls are supported on a single spreadsheet. This limit is removed in the new version of Google Sheets. In the new version of Google Sheets, spreadsheets must be explicitly granted permission to pull data from other spreadsheets using IMPORTRANGE. The first time the destination sheet pulls data from a new source sheet, the user will be prompted to grant permission. Once access is granted, any editor on the destination spreadsheet can use IMPORTRANGE to pull from any part of the source spreadsheet. The access remains in effect until the user who granted access is removed from the source.
If you can't find the key, you can use the URL of the sheet you are referencing. If I misinterpreted your question, let me know.