Adding the two lowest scores of 3 columns omitting zero

Hello, first, I apologize for probably this easy question. I am starting to use excel more and more. But, this is a scenario on the personal side.

My daughter is a skier and the cup standings take the sum of the two lowest times out of a possible 3 times. Initially all of the competitors were doing all 3 runs. They MUST complete two to be considered in the overall standings. Previously, I used the following formula to calculate the sum of the two lowest times.

=SUM(SMALL(B61:D61,{1,2}))

However, this late in the season, some racers are focused on just winning the day, so they only score once of the 3 possible runs. This is considered for the best time (lowest/fastest) for ONLY THAT DAY. But, for the overall Cup victory, assuming that the skier has completed the mandatory two runs out of 3 in each of the 3 races, they add the two together for the sum of their two lowest times.

THE QUESTION

PART 1: How do I add the two lowest scores in the range (B106:D106) and exclude any blank fields or zeros? I can go in an enter zeros for the kids that did not compete or only did 1 run if that is easiest.
PART 2: How do I add that result to a running number from a previous range (B61,D61) and (B17,D17) [3 different races]
PART 3: How do I rank the final number from PART 2 above from lowest to highest.

I struggle nesting the different functions and I always get errors saying that I have exceeded the number of functions or syntax or something about using a apostrophe instead of equal sign.

My 10 year old daughter and I thank you in advance!!!

Where will I find your mocked up expected results?

Not only are there no expected results... there are no data at all in the file!!

Scroll up, Glenn.

Oh... pooh!

To be honest, that does not help. The description does not match the file. You need to make sure the description relates to your file DIRECTLY.

Do you get Dragon's Den in USA?

You have 30-60 seconds to get my attention. If you haven't made it clear enough for me to understand the principle of what you want... I'm out and I move on.

They do, I believe, and it's called 'Sharks' Tank'.