If you want to sound like a native, you need to change all the verb tenses. It should be "One day the students were having a painting lesson. Everyone began to draw a picture...
I noticed the same thing. This is something I've noticed with my Chinese colleagues; their English tends to avoid the past tense saying "I draw the cat" instead of "I drew the cat". The first phrase would only apply during the act of drawing before its completion.
If you want to nitpick, while you're drawing the cat, you would say "I'm drawing the cat." You would only use the plain "I draw the cat" if you're talking about something you do regularly. I draw the cat every Thursday.
