To answer the first one I actually looked up the definition of the dangling structure. According to it, the answer is no. However, the form is strange indeed. I looked up why and this is what I discovered: "A present participle is most commonly used as part of the continuous form of a verb, after verbs of perception, after verbs of movement, or as an adjective". Here there is no verb of perception (see, watch, hear, etc..), nor there is any sence of continuity, as the top is apparently bought already. So it is better to change it to something simpler, like "that costed me"
To the second one - no, there could be as many verbs in a sentence as one likes. In theory :D On practice it could descend into an unreadable mess quite fast. Here everything is fine except for "say" should be in past form as well, as it is obviously a sequence of movements.
Number three - I would say something like "She was teaching me math that I already new, but I pretented not to know it to follow her teaching" But it can be reformed in many ways, so feel free to make changes :)