Are the quests offered up in sequential order in the game or do some of them come up at random? Also, what response at the initial interaction with Lola prompts further action? I've played a bit and gotten some of the later quests without having seen options for some the middle quests and wondering if I can pick them up or have to restart to find them.

Some quests require you to do other quests first, but there are several quest 'lines' that are interwoven, and the numbering is just a coinvent labeling, not actual order.