Design of A Counter Circuit
You say you are not allowed to use counter ICs..... How about making a counter IC? 

Counters are made up of flip flops in series with each other. Flip flops can be made with logic gates, or if you are not allowed to use them either, they can be made with transistors. This would solve your LED sequence problem. 

As for being able to do the whole thing during one high pulse.... you would have to build a small circuit that would oscillate 7 times during one pulse width to trigger your home made counter.