What a nice and curious problem. It's rather startling at first that the answers needn't be integers.

I don't even know that 5/3 is the best answer for the example above,

It seems most implausible that it isn't. Hmm, yes, let's see: in a better solution, each 5 stick must be cut into at most 2 pieces, obviously. The resulting (wlog) 14 pieces must be assembled into five 7-sticks so one 7-stick must get at most two pieces. One of these must be at least 3.5 long. Whichever 5-stick that came from, you had a length of 1.5 left over, which is too short.

Yes, the non-integer answer was startling to me too, and is really what made the problem stick in my head rather than falling out again shortly after it occurred to me.

The resulting (wlog) 14 pieces must be assembled into five 7-sticks so one 7-stick must get at most two pieces. One of these must be at least 3.5 long.

Oh yes, that's a nice approach to proving optimality. Thank you!

Just to point out - the 'wlog 14' is because if any 5-sticks are left whole by the dissection, we can chop them in half to get 14 pieces. I didn't make this very explicit but we need it for the final step, where the two-piece 7-stick must not be 5+2.