Average of every two cells

Hello,
I've read a lot but somehow I can't figure out why this supposedly simple task is making me so much head ache.
I have two ptoblems:
1. I need to average every two cells from one sheet into a cells of another sheet
e.g.
sheet 1
13 1.2
14 2.1
15 5.7
16 6.5
17 4.2
18 8.1
19 empty
20 45.2


From sheet A row C13 and C14 need to be averaged, then C15+C16, the C17+C18 etc and written in cells c13 in sheet B. Sometime there might be a gap with no values in.

2. I need to have the names written in column B13 from sheet A also written into sheet B in front of the calculated average
What I found is this formula:
=INDEX('Sheet B'!$B$12:$B$1001,ROWS(B13:B$13)*2) for B13
=INDEX('Sheet B'!$B$12:$B$1001,ROWS(B$13:B88)*2) for B88

Is there an easier version of that and a way to fill gaps in case no name is written in it?
I mean if I see an error I could fix it, but if it is automatically it would be wonderful.

Heck, that simple task got quite complicated.
Thank you a lot! It works!

Now, if I like to apply the same to columns D, E, F... and have the row going to infinite I changed the formula to following:
=AVERAGE(INDEX('Sheet B'!C$13:C$10000, 2*ROWS('Sheet B'!C$13:C13) - 1):INDEX('Sheet B'!C$13:C$10000,2*ROWS('Sheet B'!C$13:C13)))
I got rid off some $ and changed value 22 to 10000 (which is not infinite but a high number). I also changed C14 to C13 so the first cell is written in and not the 2nd.

I wonder if something is off so that I won't get surprises in the middle of the sheet.

thank you so much