How can I create a long (10 second) pulse from a short active pulse?
In the digital domain, a counter will be the simplest and most correct approach. Dividing a clock limits you to measuring exact binary multiples of the original clock; a counter can be configured to count up (or down) to any value.

Also, your math is way off. 10 seconds at 50 MHz comes out to 5&times;108 cycles, which can be counted in just under 29 bits (229 = 5.36&times;108). 114 DFFs would give you a counter with a period of 4.15&times;1026 seconds, well beyond the expected lifetime of the universe.