Came into this late, but being a left-field thinker I came up with a different solution to the original second puzzle before I looked at the answers. Not only that, but I only need one fuse
Cut a fuse into 60 equal pieces. Light all 60. When 45 of them finish burning, your cake is done.
You can double average accuracy by using both fuses and doubling all the numbers (but the worst-case error bounds are the same).
If you only need accuracy to within 5 minutes, then 12 pieces will do.