I've been running a morino since like December. You feed it candy as soon as the option appears, which is the Imotchi stage. The type of cocoon you get depends on the final weight and if you fed it candy all the time.
One thing I do know is if you keep the weight as low as possible (about 20-29mg iirc) you will always get Minotchi/Funkorogotchi. 40-49mg is Koganetchi/Tentotchi/Helmetchi/FutagoAritchi.
Getting Helmetchi is easy if you just feed it candy every time, get to 40-49mg, keep the temp warm. By warm I mean, consistently monitor it every 2-3 hours. Keep it paused over night if you have to, it won't penalize you. Don't let it stay at max on the hot spectrum for too long and don't let it dip past the middle for too long as well. He'll come out the cocoon. Now keep the weight at exactly 30mg. I haven't tested if it's 30-39mg, yet. But, keeping it at exactly 30mg is fine. I still fed him only candy at this point, but not sure if it matters. Once he goes back in the cocoon 2 days later just keep the temp really low. Not totally maxed out, but a tick or two off.
FutagoAritchi is essentially the same process. Feed only candy, get to 40-49mg, but keep temp super high. Almost maxed out basically the entire time, but once again monitor it. Don't let it stay max forever or it will die in the cocoon.
You can actually just feed it and wait until he beeps for you to play with him or to scare or a predator and it doesn't appear to penalize you on the cocoon you get. Using this method, I have been able to get him to 40mg by day 2. Though, if you miss too many calls for a predator he will die. This appears to be about 3 or 4 times. And with an adult I've noticed if you missed it about 2 times that's it. I've never gotten one to live past 9 days, yet.
Hope you enjoy your Morino! It's my favourite out of all the various tama's that I own.
Attached is a growth chart, however I have seen either a thread or a website that actually stated the weight levels typical to get each character. Sadly, I don't remember where I found it. Maybe someone else out there reading this knows.