First things first, make sure that you are actually meeting the initiative requirement of the challenge (consensus is generally that this means at least one ninja on your team must have initiative higher than what it is required). If you are not meeting that requirement, than it is well-known and has been repeatedly brought up in forums that the fights will be all but impossible, with similar things like what you are describing happening.

On the other hand, if you are meeting the requirement/already knew about this (in which case sorry for hashing out the details :P) and this is still happening, then we have an actual problem on our hands.

It's not the initiative requirement in this particular instance.

If you have access to the newest Speed Trials, check them out and you'll see what I mean.

Completed the new levels the day they came out, found nothing strange or broken. Check to make sure you are meeting the minimum suggested power/initiative, it DOES matter.

Pretty sure it is. Post your fastest init, and the level you are currently stuck on. I auto'd through all of the levels already, except the last one for each because the auto battle feature stopped at that one for some reason. And lots of people already finished them as well. So I doubt there is any major bug. Maybe there is some sort of minor one though? Doubt it though.

Correct

There's no bug with the new Speed Trial.

And yes, it is Ini based.

Hope that draws less confusion.