Olys are unlikely to help with the start of your sprints. Even the speed of an Olympic lift can't come close to the speed of contraction at the sound of the gun or on each stride. Heavy deadlifts will increase your strength and your sprinting will help you convert that extra strength to power. I find squats destroy my ability to sprint because they make me so sore.
Very short distances. Very long rests. Not even approaching 100m sprints. Stick to 20-40m out of blocks. I rarely do more than 6-8 reps, about 5 minutes between. No more than 10 reps in a session. This is very taxing CNS work. FYI...you're training acceleration, not speed (just a note on your title). No sprint work on back-to-back days.
If you are using blocks, here's something that helped me get out of the blocks quicker, to the tune of 0.2 second 100m improvement in the course of 2 weeks. Push back into the blocks with BOTH feet instead of just the front one. Get a quick stretch reflex on your rear leg (you'll load the calf, Achille's, and hamstring (slightly)) and then quickly turn it back over. Doing this, I feel like I'm coming out of the blocks with a rocket up my ass.
Usain Bolt demonstrating
What times are you running in the 100 and 200?