Let's hope they sell you a controller or maybe give you one. The whole kits go for $150. Controller like theirs are like $30 from China.
As for the battery, when the light turns green, let it stay on the charger.
A little education. Your battery is probably 30 or 40 small round cells, arranged 10x3 or 10 x4. The put three or four cells in parallel, Than they put ten of these parallel groups in series. It's like two 1,5 V flashlight batteries adding to made 3 volts.
Difference is that instead of 1.5V, these cells charge up to 4.2 volts. At full charge your 36V pack is really at 42 volts. When discharged, these have little power at 3.0V, so your battery shuts down around 30 volts,
Lithium batteries get in trouble if charged above 4.2 volts or discharged below 2.7 volts, By trouble, they can catch on fire, So the battery has voltage monitors inside to watcj the voltage.
With ten series groups, sometimes the voltage between them gets out of whack. One might be at 4.2V. Another might be at 3.8V. That's called unbalanced. When you charge an unbalanced battery, it stops charging when the first group hits 4.2V. If other groups are low, them they cannot get to full charge and you get less power out of the battery. If you leave it it charge though, most batteries will eventually equalize that imbalance. Takes a while. Hours.
The only risk is that batteries are most at risk of catching fire when being charged, so I don't like doing that indoors. Leave the trike outside, make sure it get rained on, and give it a longer charge.