Short answer

1. Add a new sheet.
2. Add the following formulas to the new sheet

Cell A2 -> Headers

=OFFSET('Form responses 1'!$A$2,ceiling((row()-1)/5,1)-1,0)  

This could be repeated several times, one for each header column, just change the last parameter accordingly.

Cell B2 -> data

=OFFSET('Form responses 1'!$A$2,ceiling((row()-1)/5,1)-1,mod(row()-2,5)*2+1,1,2)

Fill down as necessary

Explanation

• OFFSET() is used to get the values from the source data.
• 'Form responses 1'!$A$2 is the start point to count columns and rows.
• row() is used to increment the row and column iterators.
• ceiling((row()-1)/5,1), row iterator, is used to set the row from the source data.
• mod(row()-2,5)*2+1, column iterator, is used to set the column from the source data.
• In the data formula the last parameter, indicates the number of data columns.

In case that are several "header data" columns, add 1 for each of them. I.E. if there are five "header data" columns, and six "data columns" the column iterator should be

mod(row()-2,5)*6+5

The final data formula to be added to the F2 cell is:

=OFFSET('Form responses 1'!$A$2,ceiling((row()-1)/5,1)-1,mod(row()-2,5)*6+5,1,6)

In the row and column iterators, 5 is used because there are 5 columns for "record data".

Demo

Spreadsheet

If by "add data labels for each data point" you mean "adding labels from A column to each individual point" - then that is not possible. you can add only values as a label for each data point.

As for "two independent axis" - this can be easily achieved by adding new series from another column. All the above would look like:

From there you can add trendlines or customize each individual data point by changing color, size and shape:

If you will re-shape your table a bit you could do this:

Of course, if you really want so you could add custom label per each data point manually by inserting a drawing:

demo sheets: one, two