I'm a tiny bit uncomfortable with the notion the OP is merely suffering from culture shock, since coming from an outside culture can make you aware of things locals don't notice simply because that's the way things always have been for them. I recognise everything he describes. Myself, I come from Scotland where the drinking culture is bordering on out of control: before I came to Taiwan, I didn't think that much about it, but now that I've been away for a couple of years, it seems pretty bad. I'm sure there are Taiwanese who, after returning home after a few years elsewhere, might find things back home they wish were different - like the transport culture.
I understand OP's point of view entirely, having had a carbon bicycle completely totalled last year by a guy who came racing - and I mean, racing - around a corner on the bicycle paths and straight into me. I was reminded of it taking a walk on the seaside front at Danshui the other day, when a local in racing gear and on a fancy carbon bike zipped and swerved through the late evening crowds at some considerable speed. That he didn't smack into anyone in the dark is a minor miracle. In the last week I've seen two scooter accidents just in the streets near me. Can things change? I don't know, but I certainly hope so. When I last lived here, 2008-2010, you couldn't get a decent beer to save your life. Now there's little bars serving craft beer all over the place. Back home in the UK, the cycling culture is still struggling to take off, being perceived as a predominantly white, middle-class activity, so if there's a problem, it's certainly not just here.
In the meantime, there are, as others have pointed out, areas of cycle path outside of Dadaocheng that are almost completely empty. Unfortunately, the chap who totalled my beautiful white Giant carbon Defy was the only other person on the path at that time - although judging by his speed, he clearly thought he was alone.