Yeah, its confusing!
There are basically two scoring systems. Medals, which work normally, and chaining large enemies together, which ups a multiplier (I think it goes x2, x4, x8, x16, x32, x64, someone correct me if I'm off). That multiplier is applied to everything you kill, and is attacked to a chain timer (think Ketsui, but instead of pointblanking with Shot, you are killing large enemies).
If you have medals maxed out at 10k, you will instead get 100k medals while a chain is active. Even with the 100k medals, though, it seems you still get by far the largest part of your score by killing a boss with x64 multiplier
To add time to the chain, you have to kill another large enemy. Bombing will stop the chain from dropping until the bomb ends, but it will lower your timer to 1 (which is very low, like 1/100 a second), so you want to have a large enemy to kill after you drop the bomb, but before it ends, as that will jump the timer back up. Suiciding pauses the timer from when you die, to just about when your invincibility frames from death run out. So, you can plan your suicides to extend the chain (A suicide and 3 bombs are used to patch together the chainable enemies through both phases of the first boss, to get the x64 kill and net 11m)
As for how to display the timer... yes, how do we do this?

I know you can make that information visible, but I have not figured it out yet. I read that you "Hold C and press Up any time during the game" to turn this info on, but I haven't been able to get this information to stay on screen