Sum the lowest 3 results for multiple groups and then identify which group has the lowest

Thank you for viewing my query.

I would like to be able to sum the lowest 3 results for multiple groups and then identify which group has the lowest value.
I actually then do the same for the lowest 4 and then 5 results.

I currently use a combination of macros, sorts and then either sums or a physical calculator!
Unfortunately I am under tight time pressures when I need to do these calculations with people waiting for the results.

I have watched videos, read forum posts and webpages and often think yes that's the answer but then I fail to be able to get things to work.
I have tried all manner of sum and small formulas but alas I can't work it out.

Attached a demo spreadsheet with a results tab and a calculations tab that hopefully outlines what I am trying to achieve.
I have added a notes section and listed the criteria I can think of on the first sheet.

Hopefully I have explained what I am trying t o achieve.
Thank you in advance for your time and assistance.